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(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

In this post I address two critical issues, as raised in private correspondence with researchers, which may illuminate some objections to Gödel’s reasoning and conclusions that have been raised elsewhere by Wittgenstein, Floyd, Putnam et al.:

(i) By Rosser’s reasoning, doesn’t simple consistency suffice for defining an undecidable arithmetical proposition?

(ii) Doesn’t Gödel’s undecidable formula assert its own unprovability?

NOTE: The following correspondence refers copiously to this paper that was presented in June 2015 at the workshop on Hilbert’s Epsilon and Tau in Logic, Informatics and Linguistics, University of Montpellier, France.

Subsequently, most of the cited results were detailed formally in the following paper due to appear in the December 2016 issue of ‘Cognitive Systems Research‘:

A: Doesn’t simple consistency suffice for defining Rosser’s undecidable arithmetical proposition?

You claim that the PA system is $\omega$-inconsistent, and that Gödel’s first theorem holds vacuously. But by Rosser’s result, simple consistency suffices.

Well, it does seem surprising that Rosser’s claim—that his ‘undecidable’ proposition only assumes simple consistency—has not been addressed more extensively in the literature. Number-theoretic expositions of Rosser’s proof have generally remained either implicit or sketchy (see, for instance, this post).

Note that Rosser’s proposition and reasoning involve interpretation of an existential quantifier, whilst Gödel’s proposition and reasoning only involve interpretation of a universal quantifier.

The reason why Rosser’s claim is untenable is that—in order to interpret the existential quantifier as per Hilbert’s $\epsilon$-calculus—Rosser’s argument needs to assume his Rule C (see Elliott Mendelson, Introduction to Mathematical Logic, 1964 ed., p.73), which implicitly implies that Gödel’s arithmetic P—in which Rosser’s argumentation is grounded—is $\omega$-consistent .

See, for instance, this analysis of (a) Wang’s outline of Rosser’s argument on p.5, (b) Beth’s outline of Rosser’s argument on p.6, and (c) Mendelson’s exposition of Rosser’s argument in Section 4.2 on p.8.

Moreover, the assumption is foundationally fragile, because Rule C invalidly assumes that we can introduce an ‘unspecified’ formula denoting an ‘unspecified’ numeral into PA even if the formula has not been demonstrated to be algorithmically definable in terms of the alphabet of PA.

See Theorem 8.5 and following remarks in Section 8, pp.7-8 of this paper that was presented in June 2015 at the workshop on Hilbert’s Epsilon and Tau in Logic, Informatics and Linguistics, University of Montpellier, France.

B: As I see it, rule C is only a shortcut.

As I see it, rule C is only a shortcut; it is totally eliminable. Moreover, it is part of predicate logic, not of the Peano’s arithmetic.

Assuming that Rule C is a short cut which can always be eliminated is illusory, and is tantamount to invalidly (see Corollary 8.6, p.17 of the Epsilon 2015 paper) claiming that Hilbert’s $\epsilon$ calculus is a conservative extension of the first-order predicate calculus.

Reason: Application of Rule C invalidly (see Theorem 8.5 and following remarks in Section 8, pp.7-8 of the Epsilon 2015 paper) involves introduction of a new individual constant, say $[d]$, in a first-order theory $K$ (see Mendelson 1964, p.74, I(iv)); ‘invalidly’ since Rule C does not qualify that $[d]$ must be algorithmically computable from the alphabet of $K$—which is necessary if $K$ is first-order.

Notation: We use square brackets to indicate that the expression within the brackets denotes a well-formed formula of a formal system, say $K$, that is to be viewed syntactically merely as a first-order string of $K$—i.e, one which is finitarily constructed from the alphabet of the language of $K$—without any reference to its meaning under any interpretation of $K$.

Essentially, Rule C mirrors in $K$ the intuitionistically objectionable postulation that the formula $[(\exists x)F(x)]$ of $K$ can always be interpreted as:

$F'(a)$ holds for some element $a$

in the domain of the interpretation of $K$ under which the formula $[F(x)]$ interprets as the relation $F'(x)$.

The Epsilon 2015 paper shows that this is not a valid interpretation of the formula $[(\exists x)F(x)]$ under any finitary, evidence-based, interpretation of $K$.

That, incidentally, is a consequence of the proof that PA is not $\omega$-consistent; which itself is a consequence of (Theorem 7.1, p.15, of the Epsilon 2015 paper):

Provability Theorem for PA: A PA formula $[F(x)]$ is provable if, and only if, $[F(x)]$ interprets as an arithmetical relation $F'(x)$ that is algorithmically computable as always true (see Definition 3, p.7, of the Epsilon 2015 paper) over the structure $\mathbb{N}$ of the natural numbers.

Compare with what Gödel has essentially shown in his famous 1931 paper on formally undecidable arithmetical propositions, which is that (Lemma 8.1, p.16, of the Epsilon 2015 paper):

Gödel: There is a PA formula $[R(x, p)]$—which Gödel refers to by its Gödel number $r$—which is not provable in PA, even though $[R(x, p)]$ interprets as an arithmetical relation that is algorithmically verifiable as always true (see Definition 4, p.7, of the Epsilon 2015 paper) over the structure $\mathbb{N}$ of the natural numbers.

C: If you by-pass the intuitionist objections, would all logicist and post-formalist theories hold?

If I have understood correctly, you claim that the PA system is $\omega$-inconsistent from an intuitionistic point of view? If you by-pass the intuitionist objections, would all logicist and post-formalist theories hold?

There is nothing to bypass—the first-order Peano Arithmetic PA is a formal axiomatic system which is $\omega$-inconsistent as much for an intuitionist, as it is for a realist, a finitist, a formalist, a logicist or a nominalist.

Philosophers may differ about beliefs that are essentially unverifiable; but the $\omega$-incompleteness of PA is a verifiable logical meta-theorem that none of them would dispute.

D: Isn’t Gödel’s undecidable formula $[(\forall x)R(x, p)]$—which Gödel refers to by its Gödel number $17Gen\ r$—self-referential?

Isn’t Gödel’s undecidable formula $[(\forall x)R(x, p)]$—which Gödel refers to by its Gödel number $17Gen\ r$—self-referential and covertly paradoxical?

According to Wittgenstein it interprets in any model as a sentence that is devoid of sense, or even meaning. I think a good reason for this is that the formula is simply syntactically wrongly formed: the provability of provability is not defined and can not be consistently defined.

What you propose may be correct, but for automation systems of deduction wouldn’t $\omega$-inconsistency be much more problematic than undecidability?

How would you feel if a syntax rule is proposed, that formulas containing numerals are instantiations of open formulas that may not be part of the canonical language? Too daring, may be?

Let me briefly respond to the interesting points that you have raised.

1. The $\omega$-inconsistency of PA is a meta-theorem; it is a Corollary of the Provability Theorem of PA (Theorem 7.1, p.15, of the Epsilon 2015 paper).

2. Gödel’s PA-formula $[(\forall x)R(x, p)]$ is not an undecidable formula of PA. It is merely unprovable in PA.

3. Moreover, Gödel’s PA-formula $[\neg(\forall x)R(x, p)]$ is provable in PA, which is why the PA formula $[(\forall x)R(x, p)]$ is not an undecidable formula of PA.

4. Gödel’s PA-formula $[(\forall x)R(x, p)]$ is not self-referential.

5. Wittgenstein correctly believed—albeit purely on the basis of philosophical considerations unrelated to whether or not Gödel’s formal reasoning was correct—that Gödel was wrong in stating that the PA formula $[(\forall x)R(x, p)]$ asserts its own unprovability in PA.

Reason: We have for Gödel’s primitive recursive relation $Q(x, y)$ that:

$Q(x, p)$ is true if, and only if, the PA formula $[R(x, p)]$ is provable in PA.

However, in order to conclude that the PA formula $[(\forall x)R(x, p)]$ asserts its own unprovability in PA, Gödel’s argument must further imply—which it does not—that:

$(\forall x)Q(x, p)$ is true (and so, by Gödel’s definition of $Q(x, y)$, the PA formula $[(\forall x)R(x, p)]$ is not provable in PA) if, and only if, the PA formula $[(\forall x)R(x, p)]$ is provable in PA.

In other words, for the PA formula $[(\forall x)R(x, p)]$ to assert its own unprovability in PA, Gödel must show—which his own argument shows is impossible, since the PA formula $[(\forall x)R(x, p)]$ is not provable in PA—that:

The primitive recursive relation $Q(x, p)$ is algorithmically computable as always true if, and only if, the arithmetical relation $R'(x, p)$ is algorithmically computable as always true (where $R'(x, p)$ is the arithmetical interpretation of the PA formula $[R(x, p)]$ over the structure $\mathbb{N}$ of the natural numbers).

6. Hence, Gödel’s PA-formula $[(\forall x)R(x, p)]$ is not covertly paradoxical.

7. IF Wittgenstein believed that the PA formula $[(\forall x)R(x, p)]$ is empty of meaning and has no valid interpretation, then he was wrong, and—as Gödel justifiably believed—he could not have properly grasped Gödel’s formal reasoning that:

(i) ‘$17Gen\ r$ is not $\kappa$-provable’ is a valid meta-theorem if PA is consistent, which means that:

‘If PA is consistent and we assume that the PA formula $[(\forall x)R(x, p)]$ is provable in PA, then the PA formula $[\neg(\forall x)R(x, p)]$ must also be provable in PA; from which we may conclude that the PA formula $[(\forall x)R(x, p)]$ is not provable in PA’

(ii) ‘$Neg(17Gen\ r)$ is not $\kappa$-provable’ is a valid meta-theorem ONLY if PA is $\omega$-consistent, which means that:

‘If PA is $\omega$-consistent and we assume that the PA formula $[\neg(\forall x)R(x, p)]$ is provable in PA, then the PA formula $[(\forall x)R(x, p)]$ must also be provable in PA; from which we may conclude that the PA formula $[\neg(\forall x)R(x, p)]$ is not provable in PA’.

8. In fact the PA formula $[(\forall x)R(x, p)]$ has the following TWO meaningful interpretations (the first of which is a true arithmetical meta-statement—since the PA formula $[R(n)]$ is provable in PA for any PA-numeral $[n]$—but the second is not—since the PA formula $[(\forall x)R(x, p)]$ is not provable in PA):

(i) For any given natural number $n$, there is an algorithm which will verify that each of the arithmetical meta-statements ‘$R'(1, p)$ is true’, ‘$R'(2, p)$ is true’, …, ‘$R'(n, p)$ is true’ holds under the standard, algorithmically verifiable, interpretation $\mathbb{M}$ of PA (see \S 5, p.11 of the Epsilon 2015 paper);

(ii) There is an algorithm which will verify that, for any given natural number $n$, the arithmetical statement ‘$R'(n, p)$ is true’ holds under the finitary, algorithmically computable, interpretation $\mathbb{B}$ of PA (see \S 6, p.13 of the Epsilon 2015 paper).

9. IF Wittgenstein believed that the PA formula $[(\forall x)R(x, p)]$ is not a well-defined PA formula, then he was wrong.

Gödel’s definition of the PA formula $[(\forall x)R(x, p)]$ yields a well-formed formula in PA, and cannot be treated as ‘syntactically wrongly formed’.

10. The Provability Theorem for PA shows that both ‘proving something in PA’ and ‘proving that something is provable in PA’ are finitarily well-defined meta-mathematical concepts.

11. The Provability Theorem for PA implies that PA is complete with respect to the concepts of satisfaction, truth and provability definable in automated deduction systems, which can only define algorithmically computable truth.

12. The Provability Theorem for PA implies that PA is categorical, so you can introduce your proposed syntax rule ONLY if it leads to a conservative extension of PA.

13. Whether ‘daring’ or not, why would you want to introduce such a rule?

E: Consider these two statements of yours …

Consider these two statements of yours:

“(iv): $p$ is the Gödel-number of the formula $[(\forall x)][R(x, y)]$ of PA” and

“D(4): Gödel’s PA-formula $[(\forall x)R(x, p)]$ is not self-referential.”

If ‘$p$‘ is the Gödel-number of the open formula in para (iv), and the second argument of the closed formula $R$ in para D(4) is ‘$p$‘, then the second formula is obtained by instantiating the variable ‘$y$‘ in the first with its own Gödel-number.

So how would you call, in one word, the relation between the entire formula (in D(4)) and its second argument?

Para D(4) is an attempt to clarify precisely this point.

1. Apropos the first statement ‘(iv)’ cited by you:

From a pedantic perspective, the “relation between the entire formula (in D(4)) and its second argument” cannot be termed self-referential because the “second argument”, i.e., $p$, is the Gödel-number of the PA formula $[(\forall x)R(x, y)]$, and not that of “the entire formula (in 4)”, i.e., of the formula $[(\forall x)R(x, p)]$ itself (whose Gödel number is $17Gen\ r$).

Putting it crudely, $17Gen\ r$ is neither self-referential—nor circularly defined—because it is not defined in terms of $17Gen\ r$, but in terms of $p$.

2. Apropos the second statement ‘D(4)’ cited by you:

I would interpret:

Gödel’s PA-formula $[(\forall x)R(x, p)]$ is self-referential

to mean, in this particular context, that—as Gödel wrongly claimed:

$[(\forall x)R(x, p)]$ asserts its own unprovability in PA.

Now, if we were to accept the claim that $[(\forall x)R(x, p)]$ is self-referential in the above sense, then (as various critics of Gödel’s reasoning have pointed out) we would have to conclude further that Gödel’s argument leads to the contradiction:

$(\forall x)Q(x, p)$ is true—and so, by Gödel’s definition of $Q(x, y)$—the PA formula $[(\forall x)R(x, p)]$ is not provable in PA—if, and only if, the PA formula $[(\forall x)R(x, p)]$ is provable in PA.

However, in view of the Provability Theorem of PA (Theorem 7.1, p.15, of the Epsilon 2015 paper), this contradiction would only follow if Gödel’s argument were to establish (which it does not) that:

The primitive recursive relation $Q(x, p)$ is algorithmically computable as always true if, and only if, the arithmetical interpretation $R'(x, p)$ of the PA formula $[R(x, p)]$ is algorithmically computable as always true over the structure $\mathbb{N}$ of the natural numbers.

The reason Gödel cannot claim to have established the above is that his argument only proves the much weaker meta-statement:

The arithmetical interpretation $R'(x, p)$ of the PA formula $[R(x, p)]$ is algorithmically verifiable as always true over the structure $\mathbb{N}$ of the natural numbers.

Ergo—contrary to Gödel’s claim— Gödel’s PA-formula $[(\forall x)R(x, p)]$ is not self-referential (and so, even though Gödel’s claimed interpretation of what his own reasoning proves is wrong, there is no paradox in Gödel’s reasoning per se)!

F: Is the PA system $\omega$-inconsistent without remedy?

Is the PA system $\omega$-inconsistent without remedy? Is it possible to introduce a new axiom or new rule which by-passes the problematic unprovable statements of the Gödel-Rosser Theorems?

1. Please note that the first-order Peano Arithmetic PA is:

(i) consistent (Theorem 7.3, p.15, of the Epsilon 2015 paper); which means that for any PA-formula $[A]$, we cannot have that both $[A]$ and $[\neg A]$ are Theorems of PA;

(ii) complete (Theorem 7.1, p.15, of the Epsilon 2015 paper); which means that we cannot add an axiom to PA which is not a Theorem of PA without inviting inconsistency;

(iii) categorical (Theorem 7.2, p.15, of the Epsilon 2015 paper); which means that if $\mathbb{M}$ is an interpretation of PA over a structure $\mathbb{S}$, and $\mathbb{B}$ is an interpretation of PA over a structure $\mathbb{T}$, then $\mathbb{S}$ and $\mathbb{T}$ are identical and denote the structure $\mathbb{N}$ of the natural numbers defined by Dedekind’s axioms; and so PA has no model which contains an element that is not a natural number (see Footnote 54, p.16, of the Epsilon 2015 paper).

2. What this means with respect to Gödel’s reasoning is that:

(i) PA has no undecidable propositions, which is why it is not $\omega$-consistent (Corollary 8.4, p.16, of the Epsilon 2015 paper);

(ii) The Gödel formula $[(\forall x)R(x, p)]$ is not provable in PA; but it is algorithmically verifiable as true (Corollary 8.3, p.16, of the Epsilon 2015 paper) under the algorithmically verifiable standard interpretation $\mathbb{M}$ of PA (see Section 5, p.11, of the Epsilon 2015 paper) over the structure $\mathbb{N}$ of the natural numbers;

(iii) The Gödel formula $[(\forall x)R(x, p)]$ is not provable in PA; and it is algorithmically computable as false (Corollary 8.3, p.16, of the Epsilon 2015 paper) under the algorithmically computable finitary interpretation $\mathbb{B}$ of PA (see Section 6, p.13, of the Epsilon 2015 paper) over the structure $\mathbb{N}$ of the natural numbers;

(iv) The Gödel formula $[\neg(\forall x)R(x, p)]$ is provable in PA; and it is therefore also algorithmically verifiable as true under the algorithmically verifiable standard interpretation $\mathbb{M}$ of PA over the structure $\mathbb{N}$ of the natural numbers—which means that the logic by which the standard interpretation of PA assigns values of ‘satisfaction’ and ‘truth’ to the formulas of PA (under Tarski’s definitions) may be paraconsistent (see http://plato.stanford.edu/entries/logic-paraconsistent) since PA is consistent;

(v) The Gödel formula $[\neg(\forall x)R(x, p)]$ is provable in PA; and it is therefore algorithmically computable as true (Corollary 8.2, p.16, of the Epsilon 2015 paper) under the algorithmically computable finitary interpretation $\mathbb{B}$ of PA over the structure $\mathbb{N}$ of the natural numbers.

3. It also means that:

(a) The “Gödel-Rosser Theorem” is not a Theorem of PA;

(b) The “unprovable Gödel sentence” is not a “problematic statement”;

(c) The “PA system” does not require a “remedy” just because it is “$\omega$-inconsistent”;

(d) No “new axiom or new rule” can “by-pass the unprovable sentence”.

4. Which raises the question:

Why do you see the “unprovable Gödel sentence” as a “problematic statement” that requires a “remedy” which must “by-pass the unprovable sentence”?

Author’s working archives & abstracts of investigations

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

The Unexplained Intellect: Complexity, Time, and the Metaphysics of Embodied Thought

Christopher Mole is an associate professor of philosophy at the University of British Columbia, Vancouver. He is the author of Attention is Cognitive Unison: An Essay in Philosophical Psychology (OUP, 2011), and The Unexplained Intellect: Complexity, Time, and the Metaphysics of Embodied Thought (Routledge, 2016).

In his preface to The Unexplained Intellect, Mole emphasises that his book is an attempt to provide arguments for (amongst others) the three theses that:

(i) “Intelligence might become explicable if we treat intelligence thought as if it were some sort of computation”;

(ii) “The importance of the rapport between an organism and its environment must $\ldots$ be understood from a broadly computational perspective”;

(iii) “$\ldots$ our difficulties in accounting for our psychological orientation with respect to time are indications of the need to shift our philosophical focus away from mental states—which are altogether too static—and towards a theory of the mind in which it is dynamic mental entities that are taken to be metaphysically foundational”.

The Brains blog

Mole explains at length his main claims in The Unexplained Intellect—and the cause that those claims serve—in a lucid and penetrating, VI-part, series of invited posts in The Brains blog (a leading forum for work in the philosophy and science of mind that was founded in 2005 by Gualtiero Piccinini, and has been administered by John Schwenkler since late 2011).

In these posts, Mole seeks to make the following points.

I: The Unexplained Intellect: The mind is not a hoard of sentences

We do not currently have a satisfactory account of how minds could be had by material creatures. If such an account is to be given then every mental phenomenon will need to find a place within it. Many will be accounted for by relating them to other things that are mental, but there must come a point at which we break out of the mental domain, and account for some things that are mental by reference to some that are not. It is unclear where this break out point will be. In that sense it is unclear which mental entities are, metaphysically speaking, the most fundamental.

At some point in the twentieth century, philosophers fell into the habit of writing as if the most fundamental things in the mental domain are mental states (where these are thought of as states having objective features of the world as their truth-evaluable contents). This led to a picture in which the mind was regarded as something like a hoard of sentences. The philosophers and cognitive scientists who have operated with this picture have taken their job to be telling us what sort of content these mental sentences have, how that content is structured, how the sentences come to have it, how they get put into and taken out of storage, how they interact with one another, how they influence behaviour, and so on.

This emphasis on states has caused us to underestimate the importance of non-static mental entities, such as inferences, actions, and encounters with the world. If we take these dynamic entities to be among the most fundamental of the items in the mental domain, then — I argue — we can avoid a number of philosophical problems. Most importantly, we can avoid a picture in which intelligent thought would be beyond the capacities of any physically implementable system.

II: The Unexplained Intellect: Computation and the explanation of intelligence

A lot of philosophers think that consciousness is what makes the mind/body problem interesting, perhaps because they think that consciousness is the only part of that problem that remains wholly philosophical. Other aspects of the mind are taken to be explicable by scientific means, even if explanatorily adequate theories of them remain to be specified.

$\ldots$ I’ll remind the reader of computability theory’s power, with a view to indicating how it is that the discoveries of theoretical computer scientists place constraints on our understanding of what intelligence is, and of how it is possible.

III: The Unexplained Intellect: The importance of computability

If we found that we had been conceiving of intelligence in such a way that intelligence could not be modelled by a Turing Machine, our response should not be to conclude that some alternative must be found to a ‘Classically Computational Theory of the Mind’. To think only that would be to underestimate the scope of the theory of computability. We should instead conclude that, on the conception in question, intelligence would (be) absolutely inexplicable. This need to avoid making intelligence inexplicable places constraints on our conception of what intelligence is.

IV: The Unexplained Intellect: Consequences of imperfection

The lesson to be drawn is that, if we think of intelligence as involving the maintenance of satisfiable beliefs, and if we think of our beliefs as corresponding to a set of representational states, then our intelligence would depend on a run of good luck the chances of which are unknown.

My suggestion is that we can reach a more explanatorily satisfactory conception of intelligence if we adopt a dynamic picture of the mind’s metaphysical foundations.

V: The Unexplained Intellect: The importance of rapport

I suggest that something roughly similar is true of us. We are not guaranteed to have satisfiable beliefs, and sometimes we are rather bad at avoiding unsatisfiability, but such intelligence as we have is to be explained by reference to the rapport between our minds and the world.

Rather than starting from a set of belief states, and then supposing that there is some internal process operating on these states that enables us to update our beliefs rationally, we should start out by accounting for the dynamic processes through which the world is epistemically encountered. Much as the three-colourable map generator reliably produces three-colourable maps because it is essential to his map-making procedure that borders appear only where they will allow for three colorability, so it is essential to what it is for a state to be a belief that beliefs will appear only if there is some rapport between the believer and the world. And this rapport — rather than any internal processing considered in isolation from it — can explain the tendency for our beliefs to respect the demands of intelligence.

VI: The Unexplained Intellect: The mind’s dynamic foundations

$\ldots$ memory is essentially a form of epistemic retentiveness: One’s present knowledge counts as an instance of memory when and only when it was attained on the basis of an epistemic encounter that lies in one’s past. One can epistemically encounter a proposition as the conclusion of an argument, and so can encounter it before the occurrence of any event to which it pertains, but one cannot encounter an event in that way. In the resulting explanation of memory’s temporal asymmetry, it is the dynamic events of epistemic encountering to which we must make reference. These encounters, and not the knowledge states to which they lead, do the lion’s share of the explanatory work.

A: Simplifying Mole’s perspective

It may help simplify Mole’s thought-provoking perspective if we make an arbitrary distinction between:

(i) The mind of an applied scientist, whose primary concern is our sensory observations of a ‘common’ external world;

(ii) The mind of a philosopher, whose primary concern is abstracting a coherent perspective of the external world from our sensory observations; and

(iii) The mind of a mathematician, whose primary concern is adequately expressing such abstractions in a formal language of unambiguous communication.

My understanding of Mole’s thesis, then, is that:

(a) although a mathematician’s mind may be capable of defining the ‘truth’ value of some logical and mathematical propositions without reference to the external world,

(b) the ‘truth’ value of any logical or mathematical proposition that purports to represent any aspect of the real world must be capable of being evidenced objectively to the mind of an applied scientist; and that,

(c) of the latter ‘truths’, what should interest the mind of a philosopher is whether there are some that are ‘knowable’ completely independently of the passage of time, and some that are ‘knowable’ only partially, or incrementally, with the passage of time.

B. Support for Mole’s thesis

It also seems to me that Mole’s thesis implicitly subsumes, or at the very least echoes, the belief expressed by Chetan R. Murthy (‘An Evaluation Semantics for Classical Proofs‘, Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, 1991; also Cornell TR 91-1213):

“It is by now folklore … that one can view the values of a simple functional language as specifying evidence for propositions in a constructive logic …”

If so, the thesis seems significantly supported by the following paper that is due to appear in the December 2016 issue of ‘Cognitive Systems Research’:

The CSR paper implicitly suggests that there are, indeed, (only?) two ways of assigning ‘true’ or ‘false’ values to any mathematical description of real-world events.

C. Algorithmic computability

First, a number theoretical relation $F(x)$ is algorithmically computable if, and only if, there is an algorithm $AL_{F}$ that can provide objective evidence (cf. ibid Murthy 91) for deciding the truth/falsity of each proposition in the denumerable sequence $\{F(1), F(2), \ldots\}$.

(We note that the concept of algorithmic computability’ is essentially an expression of the more rigorously defined concept of realizability’ on p.503 of Stephen Cole Kleene’s ‘Introduction to Metamathematics‘, North Holland Publishing Company, Amsterdam.)

D. Algorithmic verifiability

Second, a number-theoretical relation $F(x)$ is algorithmically verifiable if, and only if, for any given natural number $n$, there is an algorithm $AL_{(F,\ n)}$ which can provide objective evidence for deciding the truth/falsity of each proposition in the finite sequence $\{F(1), F(2), \ldots, F(n)\}$.

We note that algorithmic computability implies the existence of an algorithm that can finitarily decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions, whereas algorithmic verifiability does not imply the existence of an algorithm that can finitarily decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions.

The following theorem (Theorem 2.1, p.37 of the CSR paper) shows that although every algorithmically computable relation is algorithmically verifiable, the converse is not true:

Theorem: There are number theoretic functions that are algorithmically verifiable but not algorithmically computable.

E. The significance of algorithmic ‘truth’ assignments for Mole’s theses

The significance of such algorithmic ‘truth’ assignments for Mole’s theses is that:

Algorithmic computability—reflecting the ambit of classical Newtonian mechanics—characterises natural phenomena that are determinate and predictable.

Such phenomena are describable by mathematical propositions that can be termed as ‘knowable completely’, since at any point of time they are algorithmically computable as ‘true’ or ‘false’.

Hence both their past and future behaviour is completely computable, and their ‘truth’ values are therefore ‘knowable’ independent of the passage of time.

Algorithmic verifiability—reflecting the ambit of Quantum mechanics—characterises natural phenomena that are determinate but unpredictable.

Such phenomena are describable by mathematical propositions that can only be termed as ‘knowable incompletely’, since at any point of time they are only algorithmically verifiable, but not algorithmically computable, as ‘true’ or ‘false’

Hence, although their past behaviour is completely computable, their future behaviour is not completely predictable, and their ‘truth’ values are not independent of the passage of time.

F. Where Mole’s implicit faith in the adequacy of set theoretical representations of natural phenomena may be misplaced

It also seems to me that, although Mole’s analysis justifiably holds that the:

$\ldots$ importance of the rapport between an organism and its environment”

has been underacknowledged, or even overlooked, by existing theories of the mind and intelligence, it does not seem to mistrust, and therefore ascribe such underacknowledgement to any lacuna in, the mathematical and epistemic foundations of the formal language in which almost all descriptions of real-world events are currently sought to be expressed, which is the language of the set theory ZF.

G. Any claim to a physically manifestable ‘truth’ must be objectively accountable

Now, so far as applied science is concerned, history teaches us that the ‘truth’ of any mathematical proposition that purports to represent any aspect of the external world must be capable of being evidenced objectively; and that such ‘truths’ must not be only of a subjective and/or revelationary nature which may require truth-certification by evolutionarily selected prophets.

(Not necessarily religious—see, for instance, Melvyn B. Nathanson’s remarks, “Desperately Seeking Mathematical Truth“, in the Opinion piece in the August 2008 Notices of the American Mathematical Society, Vol. 55, Issue 7.)

The broader significance of seeking objective accountability is that it admits the following (admittedly iconoclastic) distinction between the two fundamental mathematical languages:

1. The first-order Peano Arithmetic PA as the language of science; and

2. The first-order Set Theory ZF as the language of science fiction.

It is a distinction that is faintly reflected in Stephen G. Simpson’s more conservative perspective in his paper ‘Partial Realizations of Hilbert’s Program‘ (#6.4, p.15):

“Finitistic reasoning (my read: ‘First-order Peano Arithmetic PA’) is unique because of its clear real-world meaning and its indispensability for all scientific thought. Nonfinitistic reasoning (my read: ‘First-order Set Theory ZF’) can be accused of referring not to anything in reality but only to arbitrary mental constructions. Hence nonfinitistic mathematics can be accused of being not science but merely a mental game played for the amusement of mathematicians.”

The distinction is supported by the formal argument (detailed in the above-cited CSR paper) that:

(i) PA has two, hitherto unsuspected, evidence-based interpretations, the first of which can be treated as circumscribing the ambit of human reasoning about ‘true’ arithmetical propositions; and the second can be treated as circumscribing the ambit of mechanistic reasoning about ‘true’ arithmetical propositions.

What this means is that the language of arithmetic—formally expressed as PA—can provide all the foundational needs for all practical applications of mathematics in the physical sciences. This was was the point that I sought to make—in a limited way, with respect to quantum phenomena—in the following paper presented at Unilog 2015, Istanbul last year:

(Presented on 26’th June at the workshop on ‘Emergent Computational Logics’ at UNILOG’2015, 5th World Congress and School on Universal Logic, 20th June 2015 – 30th June 2015, Istanbul, Turkey.)

(ii) Since ZF axiomatically postulates the existence of an infinite set that cannot be evidenced (and which cannot be introduced as a constant into PA, or as an element into the domain of any interpretation of PA, without inviting inconsistency—see Theorem 1 in $\S$4 of this post), it can have no evidence-based interpretation that could be treated as circumscribing the ambit of either human reasoning about ‘true’ set-theoretical propositions, or that of mechanistic reasoning about ‘true’ set-theoretical propositions.

The language of set theory—formally expressed as ZF—thus provides the foundation for abstract structures that—although of possible interest to philosophers of science—are only mentally conceivable by mathematicians subjectively, and have no verifiable physical counterparts, or immediately practical applications of mathematics, that can materially impact on the study of physical phenomena.

The significance of this distinction can be expressed more vividly in Russell’s phraseology as:

(iii) In the first-order Peano Arithmetic PA we always know what we are talking about, even though we may not always know whether it is true or not;

(iv) In the first-order Set Theory we never know what we are talking about, so the question of whether or not it is true is only of fictional interest.

H. The importance of Mole’s ‘rapport’

Accordingly, I see it as axiomatic that the relationship between an evidence-based mathematical language and the physical phenomena that it purports to describe, must be in what Mole terms as ‘rapport’, if we view mathematics as a set of linguistic tools that have evolved:

(a) to adequately abstract and precisely express through human reasoning our observations of physical phenomena in the world in which we live and work; and

(b) unambiguously communicate such abstractions and their expression to others through objectively evidenced reasoning in order to function to the maximum of our co-operative potential in acieving a better understanding of physical phenomena.

This is the perspective that I sought to make in the following paper presented at Epsilon 2015, Montpellier, last June, where I argue against the introduction of ‘unspecifiable’ elements (such as completed infinities) into either a formal language or any of its evidence-based interpretations (in support of the argument that since a completed infinity cannot be evidence-based, it must therefore be dispensible in any purported description of reality):

(Presented on 10th June at the Epsilon 2015 workshop on ‘Hilbert’s Epsilon and Tau in Logic, Informatics and Linguistics’, 10th June 2015 – 12th June 2015, University of Montpellier, France.)

I. Why mathematical reasoning must reflect an ‘agnostic’ perspective

Moreover, from a non-mathematician’s perspective, a Propertarian like Curt Doolittle would seem justified in his critique (comment of June 2, 2016 in this Quanta review) of the seemingly ‘mystical’ and ‘irrelevant’ direction in which conventional interpretations of Hilbert’s ‘theistic’ and Brouwer’s ‘atheistic’ reasoning appear to have pointed mainstream mathematics for, as I argue informally in an earlier post, the ‘truths’ of any mathematical reasoning must reflect an ‘agnostic’ perspective.

Author’s working archives & abstracts of investigations

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

In a recent paper A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory, authors Adam Yedidia and Scott Aaronson argue upfront in their Introduction that:

Like any axiomatic system capable of encoding arithmetic, ZFC is constrained by Gödel’s two incompleteness theorems. The first incompleteness theorem states that if ZFC is consistent (it never proves both a statement and its opposite), then ZFC cannot also be complete (able to prove every true statement). The second incompleteness theorem states that if ZFC is consistent, then ZFC cannot prove its own consistency. Because we have built modern mathematics on top of ZFC, we can reasonably be said to have assumed ZFC’s consistency.

The question arises:

How reasonable is it to build modern mathematics on top of a Set Theory such as ZF?

Some immediate points to ponder upon (see also reservations expressed by Stephen G. Simpson in Logic and Mathematics and in Partial Realizations of Hilbert’s Program):

1. “Like any axiomatic system capable of encoding arithmetic, …”

The implicit assumption here that every ZF formula which is provable about the finite ZF ordinals must necessarily interpret as a true proposition about the natural numbers is fragile since, without such an assumption, we can only conclude from Goodstein’s argument (see Theorem 1.1 here) that a Goodstein sequence defined over the finite ZF ordinals must terminate even if the corresponding Goodstein sequence over the natural numbers does not terminate!

2. “ZFC is constrained by Gödel’s two incompleteness theorems. The first incompleteness theorem states that if ZFC is consistent (it never proves both a statement and its opposite), then ZFC cannot also be complete (able to prove every true statement). The second incompleteness theorem states that if ZFC is consistent, then ZFC cannot prove its own consistency.”

The implicit assumption here is that ZF is $\omega$-consistent, which implies that ZF is consistent and must therefore have an interpretation over some mathematically definable structure in which ZF theorems interpret as ‘true’.

The question arises: Must such ‘truth’ be capable of being evidenced objectively, or is it only of a subjective, revelationary, nature (which may require truth-certification by evolutionarily selected prophets—see Nathanson’s remarks as cited in this post)?

The significance of seeking objective accountbility is that in a paper, “The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas’ Gödelian Thesis“, which is due to appear in the December 2016 issue of Cognitive Systems Research, we show (see also this post) that the first-order Peano Arithmetic PA:

(i) is finitarily consistent; but

(ii) is not $\omega$-consistent; and

(iii) has no ‘undecidable’ arithmetical proposition (whence both of Gödel’s Incompleteness Theorems hold vacuously so far as the arithmetic of the natural numbers is concerned).

3. “Because we have built modern mathematics on top of ZFC, we can reasonably be said to have assumed ZFC’s consistency.”

Now, one justification for such an assumption (without which it may be difficult to justify building modern mathematics on top of ZF) could be the belief that acquisition of set-theoretical knowledge by students of mathematics has some essential educational dimension.

If so, one should take into account not only the motivations of such a student for the learning of mathematics, but also those of a mathematician for teaching it.

This, in turn, means that both the content of the mathematics which is to be learnt (or taught), as well as the putative utility of such learning (or teaching) for a student (or teacher), merit consideration.

Considering content, I would iconoclastically submit that the least one may then need to accomodate is the following distinction between the two fundamental mathematical languages:

1. The first-order Peano Arithmetic PA, which is the language of science; and

2. The first-order Set Theory ZF, which is the language of science fiction.

A distinction that is reflected in Stephen G. Simpson’s more conservative perspective in Partial Realizations of Hilbert’s Program ($\S$6.4, p.15):

Finitistic reasoning (read ‘First-order Peano Arithmetic PA’) is unique because of its clear real-world meaning and its indispensability for all scientific thought. Nonfinitistic reasoning (read ‘First-order Set Thyeory ZF’) can be accused of referring not to anything in reality but only to arbitrary mental constructions. Hence nonfinitistic mathematics can be accused of being not science but merely a mental game played for the amusement of mathematicians.

Reason:

(i) PA has two, hitherto unsuspected, evidence-based interpretations (see this post), the first of which can be treated as circumscribing the ambit of human reasoning about true’ arithmetical propositions; and the second can be treated as circumscribing the ambit of mechanistic reasoning about true’ arithmetical propositions.

It is this language of arithmetic—formally expressed as PA—that provides the foundation for all practical applications of mathematics where the latter could be argued as having an essential educational dimension.

(ii) Since ZF axiomatically postulates the existence of an infinite set that cannot be evidenced (and which cannot be introduced as a constant into PA, or as an element into the domain of any interpretation of PA, without inviting inconsistency—see paragraph 4.2 of this post), it can have no evidence-based interpretation that could be treated as circumscribing the ambit of either human reasoning about true’ set-theoretical propositions, or that of mechanistic reasoning about true’ set-theoretical propositions.

The language of set theory—formally expressed as ZF—thus provides the foundation for abstract structures that are only mentally conceivable by mathematicians (subjectively?), and have no physical counterparts, or immediately practical applications of mathematics, which could meaningfully be argued as having an essential educational dimension.

The significance of this distinction can be expressed more vividly in Russell’s phraseology as:

(iii) In the first-order Peano Arithmetic PA we always know what we are talking about, even though we may not always know whether it is true or not;

(iv) In the first-order Set Theory we never know what we are talking about, so the question of whether or not it is true is only of fictional interest.

The distinction is lost when—as seems to be the case currently—we treat the acquisition of mathematical knowledge as necessarily including the body of essentially set-theoretic theorems—to the detriment, I would argue, of the larger body of aspiring students of mathematics whose flagging interest in acquiring such a wider knowledge in universities around the world reflects the fact that, for most students, their interests seem to lie primarily in how a study of mathematics can enable them to:

(a) adequately abstract and precisely express through human reasoning their experiences of the world in which they live and work; and

(b) unambiguously communicate such abstractions and their expression to others through objectively evidenced reasoning in order to function to the maximum of their latent potential in acieving their personal real-world goals.

In other words, it is not obvious how how any study of mathematics that has the limited goals (a) and (b) can have any essentially educational dimension that justifies the assumption that ZF is consistent.

Author’s working archives & abstracts of investigations

A foundational argument for defining Effective Computability formally, and weakening the Church and Turing Theses – II

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

$\S 1$ The Logical Issue

In the previous posts we addressed first the computational issue, and second the philosophical issue—concerning the informal concept of effective computability’—that seemed implicit in Selmer Bringsjord’s narrational case against Church’s Thesis [1].

We now address the logical issue that leads to a formal definability of this concept which—arguably—captures our intuitive notion of the concept more fully.

We note that in this paper on undecidable arithmetical propositions we have shown how it follows from Theorem VII of Gödel’s seminal 1931 paper that every recursive function $f(x_{1}, x_{2})$ is representable in the first-order Peano Arithmetic PA by a formula $[F(x_{1}, x_{2}, x_{3})]$ which is algorithmically verifiable, but not algorithmically computable, if we assume (Aristotle’s particularisation) that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counter-example under the standard interpretation of PA.

In this earlier post on the Birmingham paper, we have also shown that:

We shall argue in this post that the standard postulation of the Church-Turing Thesis—which postulates that the intuitive concept of effective computability’ is completely captured by the formal notion of algorithmic computability’—does not hold if we formally define a number-theoretic formula as effectively computable if, and only if, it is algorithmically verifiable; and it therefore needs to be replaced by a weaker postulation of the Thesis as an instantiational equivalence.

$\S 2$ Weakening the Church and Turing Theses

We begin by noting that the following theses are classically equivalent [1]:

Standard Church’s Thesis: [2] A number-theoretic function (or relation, treated as a Boolean function) is effectively computable if, and only if, it is recursive [3].

Standard Turing’s Thesis: [4] A number-theoretic function (or relation, treated as a Boolean function) is effectively computable if, and only if, it is Turing-computable [5].

In this paper we shall argue that, from a foundational perspective, the principle of Occam’s razor suggests the Theses should be postulated minimally as the following equivalences:

Weak Church’s Thesis: A number-theoretic function (or relation, treated as a Boolean function) is effectively computable if, and only if, it is instantiationally equivalent to a recursive function (or relation, treated as a Boolean function).

Weak Turing’s Thesis: A number-theoretic function (or relation, treated as a Boolean function) is effectively computable if, and only if, it is instantiationally equivalent to a Turing-computable function (or relation, treated as a Boolean function).

$\S 2.1$ The need for explicitly distinguishing between instantiational’ and uniform’ methods

Why Church’s Thesis?

It is significant that both Kurt Gödel (initially) and Alonzo Church (subsequently—possibly under the influence of Gödel’s disquietitude) enunciated Church’s formulation of effective computability’ as a Thesis because Gödel was instinctively uncomfortable with accepting it as a definition that minimally captures the essence of intuitive effective computability’ [6].

Kurt Gödel’s reservations

Gödel’s reservations seem vindicated if we accept that a number-theoretic function can be effectively computable instantiationally (in the sense of being algorithmically verifiable as defined in the Birmingham paper, reproduced in this post), but not by a uniform method (in the sense of being algorithmically computable as defined in the Birmingham paper, reproduced in this post).

The significance of the fact (considered in the Birmingham paper, reproduced in this post) that truth’ too can be effectively decidable both instantiationally and by a uniform method under the standard interpretation of PA is reflected in Gödel’s famous 1951 Gibbs lecture[7], where he remarks:

“I wish to point out that one may conjecture the truth of a universal proposition (for example, that I shall be able to verify a certain property for any integer given to me) and at the same time conjecture that no general proof for this fact exists. It is easy to imagine situations in which both these conjectures would be very well founded. For the first half of it, this would, for example, be the case if the proposition in question were some equation $F(n) = G(n)$ of two number-theoretical functions which could be verified up to very great numbers $n$.” [8]

Alan Turing’s perspective

Such a possibility is also implicit in Turing’s remarks [9]:

“The computable numbers do not include all (in the ordinary sense) definable numbers. Let P be a sequence whose n-th figure is 1 or 0 according as n is or is not satisfactory. It is an immediate consequence of the theorem of $\S8$ that P is not computable. It is (so far as we know at present) possible that any assigned number of figures of P can be calculated, but not by a uniform process. When sufficiently many figures of P have been calculated, an essentially new method is necessary in order to obtain more figures.”

Boolos, Burgess and Jeffrey’s query

The need for placing such a distinction on a formal basis has also been expressed explicitly on occasion [10].

Thus, Boolos, Burgess and Jeffrey [11] define a diagonal halting function, $d$, any value of which can be decided effectively, although there is no single algorithm that can effectively compute $d$.

Now, the straightforward way of expressing this phenomenon should be to say that there are well-defined number-theoretic functions that are effectively computable instantiationally but not uniformly. Yet, following Church and Turing, such functions are labeled as uncomputable [12]!

However, as Boolos, Burgess and Jeffrey note quizically:

“According to Turing’s Thesis, since $d$ is not Turing-computable, $d$ cannot be effectively computable. Why not? After all, although no Turing machine computes the function $d$, we were able to compute at least its first few values, For since, as we have noted, $f_{1} = f_{2} = f_{3} =$ the empty function we have $d(1) = d(2) = d(3) = 1$. And it may seem that we can actually compute $d(n)$ for any positive integer $n$—if we don’t run out of time.” [13]

Why should Chaitin’s constant $\Omega$ be labelled uncomputable’?

The reluctance to treat a function such as $d(n)$—or the function $\Omega(n)$ that computes the $n^{th}$ digit in the decimal expression of a Chaitin constant $\Omega$ [14]—as computable, on the grounds that the time’ needed to compute it increases monotonically with $n$, is curious [15]; the same applies to any total Turing-computable function $f(n)$![16]

Moreover, such a reluctance to treat instantiationally computable functions such as $d(n)$ as not effectively computable’ is difficult to reconcile with a conventional wisdom that holds the standard interpretation of the first order Peano Arithmetic PA as defining an intuitively sound model of PA.

Reason: We have shown in the Birmingham paper (reproduced in this post) that ‘satisfaction’ and ‘truth’ under the standard interpretation of PA is definable constructively in terms of algorithmic verifiability (instantiational computability).

$\S 2.2$ Distinguishing between algorithmic verifiability and algorithmic computability

We now show in Theorem 1 that if Aristotle’s particularisation is presumed valid over the structure $\mathbb{N}$ of the natural numbers—as is the case under the standard interpretation of the first-order Peano Arithmetic PA—then it follows from the instantiational nature of the (constructively defined [17]) Gödel $\beta$-function that a primitive recursive relation can be instantiationally equivalent to an arithmetical relation, where the former is algorithmically computable over $\mathbb{N}$, whilst the latter is algorithmically verifiable (i.e., instantiationally computable) but not algorithmically computable over $\mathbb{N}$.[18]

$\S 2.2.1$ Significance of Gödel’s $\beta$-function

We note first that in Theorem VII of his seminal 1931 paper on formally undecidable arithmetical propositions Gödel showed that, given a total number-theoretic function $f(x)$ and any natural number $n$, we can construct a primitive recursive function $\beta(z, y, x)$ and natural numbers $b_{n}, c_{n}$ such that $\beta(b_{n}, c_{n}, i)$ $= f(i)$ for all $0 \leq i \leq n$.

In this paper we shall essentially answer the following question affirmatively:

Query 3: Does Gödel’s Theorem VII admit construction of an arithmetical function $A(x)$ such that:

(a) for any given natural number $n$, there is an algorithm that can verify $A(i) = f(i)$ for all $0 \leq i \leq n$ (hence $A(x)$ may be said to be algorithmically verifiable if $f(x)$ is recursive);

(b) there is no algorithm that can verify $A(i) = f(i)$ for all $0 \leq i$ (so $A(x)$ may be said to be algorithmically uncomputable)?

$\S 2.2.2$ Defining effective computability

Now, in the Birmingham paper (reproduced in this post), we have formally defined what it means for a formula of an arithmetical language to be:

(i) Algorithmically verifiable;

(ii) Algorithmically computable.

under an interpretation.

We shall thus propose the definition:

Effective computability: A number-theoretic formula is effectively computable if, and only if, it is algorithmically verifiable.

Intuitionistically unobjectionable: We note first that since every finite set of integers is recursive, every well-defined number-theoretical formula is algorithmically verifiable, and so the above definition is intuitionistically unobjectionable; and second that the existence of an arithmetic formula that is algorithmically verifiable but not algorithmically computable (Theorem 1) supports Gödel’s reservations on Alonzo Church’s original intention to label his Thesis as a definition [19].

The concept is well-defined, since we have shown in the Birmingham paper (reproduced in this post) that the algorithmically verifiable and the algorithmically computable PA formulas are well-defined under the standard interpretation of PA and that:

(a) The PA-formulas are decidable as satisfied / unsatisfied or true / false under the standard interpretation of PA if, and only if, they are algorithmically verifiable;

(b) The algorithmically computable PA-formulas are a proper subset of the algorithmically verifiable PA-formulas;

(c) The PA-axioms are algorithmically computable as satisfied / true under the standard interpretation of PA;

(d) Generalisation and Modus Ponens preserve algorithmically computable truth under the standard interpretation of PA;

(e) The provable PA-formulas are precisely the ones that are algorithmically computable as satisfied / true under the standard interpretation of PA.

$\S 3$ Gödel’s Theorem VII and algorithmically verifiable, but not algorithmically computable, arithmetical propositions

In his seminal 1931 paper on formally undecidable arithmetical propositions, Gödel defined a curious primitive recursive function—Gödel’s $\beta$-function—as [20]:

Definition 1: $\beta (x_{1}, x_{2}, x_{3}) = rm(1+(x_{3}+ 1) \star x_{2}, x_{1})$

where $rm(x_{1}, x_{2})$ denotes the remainder obtained on dividing $x_{2}$ by $x_{1}$.

Gödel showed that the above function has the remarkable property that:

Lemma 1: For any given denumerable sequence of natural numbers, say $f(k, 0),\ f(k, 1),\ \ldots$, and any given natural number $n$, we can construct natural numbers $b, c, j$ such that:

(i) $j = max(n, f(k, 0), f(k, 1), \ldots, f(k, n))$;

(ii) $c = j$!;

(iii) $\beta(b, c, i) = f(k, i)$ for $0 \leq i \leq n$.

Proof: This is a standard result [21]. $\Box$

Now we have the standard definition [22]:

Definition 2: A number-theoretic function $f(x_{1}, \ldots, x_{n})$ is said to be representable in PA if, and only if, there is a PA formula $[F(x_{1}, \dots, x_{n+1})]$ with the free variables $[x_{1}, \ldots, x_{n+1}]$, such that, for any given natural numbers $k_{1}, \ldots, k_{n+1}$:

(i) if $f(k_{1}, \ldots, k_{n}) = k_{n+1}$ then PA proves: $[F(k_{1}, \ldots, k_{n}, k_{n+1})]$;

(ii) PA proves: $[(\exists_{1} x_{n+1})F(k_{1}, \ldots, k_{n}, x_{n+1})]$.

The function $f(x_{1}, \ldots, x_{n})$ is said to be strongly representable in PA if we further have that:

(iii) PA proves: $[(\exists_{1} x_{n+1})F(x_{1}, \ldots, x_{n}, x_{n+1})]$

Interpretation of $[\exists_{1}]$‘: The symbol $[\exists_{1}]$‘ denotes uniqueness’ under an interpretation which assumes that Aristotle’s particularisation holds in the domain of the interpretation.

Formally, however, the PA formula:

$[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$

is merely a short-hand notation for the PA formula:

$[\neg(\forall x_{3})\neg F(x_{1}, x_{2}, x_{3}) \wedge (\forall y)(\forall z)(F(x_{1}, x_{2}, y) \wedge F(x_{1}, x_{2}, z) \rightarrow y=z)]$.

We then have:

Lemma 2 $\beta(x_{1}, x_{2}, x_{3})$ is strongly represented in PA by $[Bt(x_{1}, x_{2}, x_{3}, x_{4})]$, which is defined as follows:

$[(\exists w)(x_{1} = ((1 + (x_{3} + 1)\star x_{2}) \star w + x_{4}) \wedge (x_{4} < 1 + (x_{3} + 1) \star x_{2}))]$.

Proof: This is a standard result [23]. $\Box$

Gödel further showed (also under the tacit, but critical, presumption of Aristotle’s particularisation [24] that:

Lemma 3: If $f(x_{1}, x_{2})$ is a recursive function defined by:

(i) $f(x_{1}, 0) = g(x_{1})$

(ii) $f(x_{1}, (x_{2}+1)) = h(x_{1}, x_{2}, f(x_{1}, x_{2}))$

where $g(x_{1})$ and $h(x_{1}, x_{2}, x_{3})$ are recursive functions of lower rank [25] that are represented in PA by well-formed formulas $[G(x_{1}, x_{2})]$ and $[H(x_{1}, x_{2}, x_{3}, x_{4})]$,

then $f(x_{1}, x_{2})$ is represented in PA by the following well-formed formula, denoted by $[F(x_{1}, x_{2}, x_{3})]$:

$[(\exists u)(\exists v)(((\exists w)(Bt(u, v, 0, w) \wedge G(x_{1}, w))) \wedge Bt(u, v, x_{2}, x_{3}) \wedge (\forall w)(w < x_{2} \rightarrow (\exists y)(\exists z)(Bt(u, v, w, y) \wedge Bt(u, v, (w+1), z) \wedge H(x_{1}, w, y, z)))].$

Proof: This is a standard result [26]. $\Box$

$\S 4.1$ What does “$[(\exists_{1} x_{3})F(k, m, x_{3})]$ is provable” assert under the standard interpretation of PA?

Now, if the PA formula $[F(x_{1}, x_{2}, x_{3})]$ represents in PA the recursive function denoted by $f(x_{1}, x_{2})$ then by definition, for any given numerals $[k], [m]$, the formula $[(\exists_{1} x_{3})F(k, m, x_{3})]$ is provable in PA; and true under the standard interpretation of PA.

We thus have that:

Lemma 4:$[(\exists_{1} x_{3})F(k, m, x_{3})]$ is true under the standard interpretation of PA” is the assertion that:

Given any natural numbers $k, m$, we can construct natural numbers $t_{(k, m)}, u_{(k, m)}, v_{(k, m)}$—all functions of $k, m$—such that:

(a) $\beta(u_{(k, m)}, v_{(k, m)}, 0) = g(k)$;

(b) for all $i, $\beta(u_{(k, m)}, v_{(k, m)}, i) = h(k, i, f(k, i))$;

(c) $\beta(u_{(k, m)}, v_{(k, m)}, m) = t_{(k, m)}$;

where $f(x_{1}, x_{2})$, $g(x_{1})$ and $h(x_{1}, x_{2}, x_{3})$ are any recursive functions that are formally represented in PA by $F(x_{1}, x_{2}, x_{3}), G(x_{1}, x_{2})$ and $H(x_{1}, x_{2}, x_{3}, x_{4})$ respectively such that:

(i) $f(k, 0) = g(k)$

(ii) $f(k, (y+1)) = h(k, y, f(k, y))$ for all $y

(iii) $g(x_{1})$ and $h(x_{1}, x_{2}, x_{3})$ are recursive functions that are assumed to be of lower rank than $f(x_{1}, x_{2})$.

Proof: For any given natural numbers $k$ and $m$, if $[F(x_{1}, x_{2}, x_{3})]$ interprets as a well-defined arithmetical relation under the standard interpretation of PA, then we can define a deterministic Turing machine $TM$ that can construct’ the sequences:

$f(k, 0), f(k, 1), \ldots, f(k, m)$

and:

$\beta(u_{(k, m)}, v_{(k, m)}, 0), \beta(u_{(k, m)}, v_{(k, m)}, 1), \ldots, \beta(u_{(k, m)}, v_{(k, m)}, m)$

and give evidence to verify the assertion. $\Box$[27]

We now see that:

Theorem 1: Under the standard interpretation of PA $[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$ is algorithmically verifiable, but not algorithmically computable, as always true over $\mathbb{N}$.

Proof: It follows from Lemma 4 that:

(1) $[(\exists_{1} x_{3})F(k, m, x_{3})]$ is PA-provable for any given numerals $[k, m]$. Hence $[(\exists_{1} x_{3})F(k, m, x_{3})]$ is true under the standard interpretation of PA. It then follows from the definition of $[F(x_{1}, x_{2}, x_{3})]$ in Lemma 3 that, for any given natural numbers $k, m$, we can construct some pair of natural numbers $u_{(k, m)}, v_{(k, m)}$—where $u_{(k, m)}, v_{(k, m)}$ are functions of the given natural numbers $k$ and $m$—such that:

(a) $\beta(u_{(k, m)}, v_{(k, m)}, i) = f(k, i)$ for $0 \leq i \leq m$;

(b) $F^{*}(k, m, f(k, m))$ holds in $\mathbb{N}$.

Since $\beta(x_{1}, x_{2}, x_{3})$ is primitive recursive, $\beta(u_{(k, m)}, v_{(k, m)}, i)$ defines a deterministic Turing machine $TM$ that can construct’ the denumerable sequence $f'(k, 0), f'(k, 1), \ldots$ for any given natural numbers $k$ and $m$ such that:

(c) $f(k, i) = f'(k, i)$ for $0 \leq i \leq m$.

We can thus define a deterministic Turing machine $TM$ that will give evidence that the PA formula $[(\exists_{1} x_{3})F(k, m, x_{3})]$ is true under the standard interpretation of PA.

Hence $[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$ is algorithmically verifiable over $\mathbb{N}$ under the standard interpretation of PA.

(2) Now, the pair of natural numbers $u_{(x_{1}, x_{2})}, v_{(x_{1}, x_{2})}$ are defined such that:

(a) $\beta(u_{(x_{1}, x_{2})}, v_{(x_{1}, x_{2})}, i) = f(x_{1}, i)$ for $0 \leq i \leq x_{2}$;

(b) $F^{*}(x_{1}, x_{2}, f(x_{1}, x_{2}))$ holds in $\mathbb{N}$;

where $v_{(x_{1}, x_{2})}$ is defined in Lemma 3 as $j$!, and:

(c) $j = max(n, f(x_{1}, 0), f(x_{1}, 1), \ldots, f(x_{1}, x_{2}))$;

(d) $n$ is the number’ of terms in the sequence $f(x_{1}, 0), f(x_{1}, 1), \ldots, f(x_{1}, x_{2})$.

Since $j$ is not definable for a denumerable sequence $\beta(u_{(x_{1}, x_{2})}, v_{(x_{1}, x_{2})}, i)$ we cannot define a denumerable sequence $f'(x_{1}, 0), f'(x_{1}, 1), \ldots$ such that:

(e) $f(k, i) = f'(k, i)$ for all $i \geq 0$.

We cannot thus define a deterministic Turing machine $TM$ that will give evidence that the PA formula $[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$ interprets as true under the standard interpretation of PA for any given sequence of numerals $[(a_{1}, a_{2})]$.

Hence $[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$ is not algorithmically computable over $\mathbb{N}$ under the standard interpretation of PA.

The theorem follows. $\Box$

Corollary 1: If the standard interpretation of PA is sound, then the classical Church and Turing theses are false.

The above theorem now suggests the following definition:

Definition 2: (Effective computability) A number-theoretic function is effectively computable if, and only if, it is algorithmically verifiable.

Such a definition of effective computability now allows the classical Church and Turing theses to be expressed as the weak equivalences in $\S 2$—rather than as identities—without any apparent loss of generality.

References

BBJ03 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic (4th ed). Cambridge University Press, Cambridge.

Bri93 Selmer Bringsjord. 1993. The Narrational Case Against Church’s Thesis. Easter APA meetings, Atlanta.

Ch36 Alonzo Church. 1936. An unsolvable problem of elementary number theory. In M. Davis (ed.). 1965. The Undecidable Raven Press, New York. Reprinted from the Am. J. Math., Vol. 58, pp.345-363.

Ct75 Gregory J. Chaitin. 1975. A Theory of Program Size Formally Identical to Information Theory. J. Assoc. Comput. Mach. 22 (1975), pp. 329-340.

Go31 Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York.

Go51 Kurt Gödel. 1951. Some basic theorems on the foundations of mathematics and their implications. Gibbs lecture. In Kurt Gödel, Collected Works III, pp.304-323.\ 1995. Unpublished Essays and Lectures. Solomon Feferman et al (ed.). Oxford University Press, New York.

Ka59 Laszlo Kalmár. 1959. An Argument Against the Plausibility of Church’s Thesis. In Heyting, A. (ed.) Constructivity in Mathematics. North-Holland, Amsterdam.

Kl36 Stephen Cole Kleene. 1936. General Recursive Functions of Natural Numbers. Math. Annalen vol. 112 (1936) pp.727-766.

Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton.

Me90 Elliott Mendelson. 1990. Second Thoughts About Church’s Thesis and Mathematical Proofs. Journal of Philosophy 87.5.

Pa71 Rohit Parikh. 1971. Existence and Feasibility in Arithmetic. The Journal of Symbolic Logic, Vol.36, No. 3 (Sep., 1971), pp. 494-508.

Si97 Wilfried Sieg. 1997. Step by recursive step: Church’s analysis of effective calculability Bulletin of Symbolic Logic, Volume 3, Number 2.

Sm07 Peter Smith. 2007. Church’s Thesis after 70 Years. A commentary and critical review of Church’s Thesis After 70 Years. In Meinong Studies Vol 1 (Ontos Mathematical Logic 1), 2006 (2013), Eds. Adam Olszewski, Jan Wolenski, Robert Janusz. Ontos Verlag (Walter de Gruyter), Frankfurt, Germany.

Tu36 Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

An12 … 2012. Evidence-Based Interpretations of PA. In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

Notes

Return to 1: cf. Me64, p.237.

Return to 2: Church’s (original) Thesis: The effectively computable number-theoretic functions are the algorithmically computable number-theoretic functions Ch36.

Return to 11: cf. Me64, p.227.

Return to 4: After describing what he meant by “computable” numbers in the opening sentence of his 1936 paper on Computable Numbers Tu36, Turing immediately expressed this thesis—albeit informally—as: “… the computable numbers include all numbers which could naturally be regarded as computable”.

Return to 5: cf. BBJ03, p.33.

Return to 6: See Si97.

Return to 7: Go51.

Return to 8: Parikh’s paper Pa71 can also be viewed as an attempt to investigate the consequences of expressing the essence of Gödel’s remarks formally.

Return to 9: Tu36, $\S9(II)$, p.139.

Return to 10: Parikh’s distinction between decidability’ and feasibility’ in Pa71 also appears to echo the need for such a distinction.

Return to 11: BBJ03, p. 37.

Return to 12: The issue here seems to be that, when using language to express the abstract objects of our individual, and common, mental concept spaces’, we use the word exists’ loosely in three senses, without making explicit distinctions between them (see An07).

Return to 13: BBJ03, p.37.

Return to 14: Chaitin’s Halting Probability is given by $0 < \Omega = \sum2^{-|p|} < 1$, where the summation is over all self-delimiting programs $p$ that halt, and $|p|$ is the size in bits of the halting program $p$; see Ct75.

Return to 15: The incongruity of this is addressed by Parikh in Pa71.

Return to 16: The only difference being that, in the latter case, we know there is a common program’ of constant length that will compute $f(n)$ for any given natural number $n$; in the former, we know we may need distinctly different programs for computing $f(n)$ for different values of $n$, where the length of the program will, sometime, reference $n$.

Return to 17: By Kurt Gödel; see Go31, Theorem VII.

Return to 18: Analagous distinctions in analysis: The distinction between algorithmically computable, and algorithmically verifiable but not algorithmically computable, number-theoretic functions seeks to reflect in arithmetic the essence of uniform methods (formally detailed in the Birmingham paper (reproduced in this post) and in its main consequence—the Provability Theorem for PA—as detailed in this post), classically characterised by the distinctions in analysis between: (a) uniformly continuous, and point-wise continuous but not uniformly continuous, functions over an interval; (b) uniformly convergent, and point-wise convergent but not uniformly convergent, series.

A limitation of set theory and a possible barrier to computation: We note, further, that the above distinction cannot be reflected within a language—such as the set theory ZF—which identifies equality’ with equivalence’. Since functions are defined extensionally as mappings, such a language cannot recognise that a set which represents a primitive recursive function may be equivalent to, but computationally different from, a set that represents an arithmetical function; where the former function is algorithmically computable over $\mathbb{N}$, whilst the latter is algorithmically verifiable but not algorithmically computable over $\mathbb{N}$.

Return to 19: See the Provability Theorem for PA in this post.

Return to 20: cf. Go31, p.31, Lemma 1; Me64, p.131, Proposition 3.21.

Return to 21: cf. Go31, p.31, Lemma 1; Me64, p.131, Proposition 3.22.

Return to 22: Me64, p.118.

Return to 23: cf. Me64, p.131, proposition 3.21.

Return to 24: The implicit assumption being that the negation of a universally quantified formula of the first-order predicate calculus is indicative of “the existence of a counter-example”—Go31, p.32.

Return to 25: cf. Me64, p.132; Go31, p.30(2).

Return to 26: cf. Go31, p.31(2); Me64}, p.132.

Return to 27: A critical philosophical issue that we do not address here is whether the PA formula $[F(x_{1}, x_{2}, x_{3}]$ can be considered to interpret under a sound interpretation of PA as a well-defined predicate, since the denumerable sequences $\{f(k, 0), f(k, 1), \ldots, f(k, m), m_{p}: p>0$ and $m_{p}$ is not equal to $m_{q}$ if $p$ is not equal to $q\}$—are represented by denumerable, distinctly different, functions $\beta(u_{p_{1}}, v_{p_{2}}, i)$ respectively. There are thus denumerable pairs $(u_{p_{1}}, v_{p_{2}})$ for which $\beta(u_{p_{1}}, v_{p_{2}}, i)$ yields the sequence $f(k, 0), f(k, 1), \ldots, f(k, m)$.

Author’s working archives & abstracts of investigations

A foundational argument for defining Effective Computability formally, and weakening the Church and Turing Theses – I

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

$\S 1.1$ The Philosophical Issue

In a previous post we have argued that standard interpretations of classical theory may inadvertently be weakening a desirable perception—of mathematics as the lingua franca of scientific expression—by ignoring the possibility that since mathematics is, indeed, indisputably accepted as the language that most effectively expresses and communicates intuitive truth, the chasm between formal truth and provability must, of necessity, be bridgeable.

We further queried whether the roots of such interpretations may lie in removable ambiguities that currently persist in the classical definitions of foundational elements; ambiguities that allow the introduction of non-constructive—hence non-verifiable, non-computational, ambiguous and essentially Platonic—elements into the standard interpretations of classical mathematics.

Query 1: Are formal classical theories essentially unable to adequately express the extent and range of human cognition, or does the problem lie in the way formal theories are classically interpreted at the moment?

We noted that the former addressed the question of whether there are absolute limits on our capacity to express human cognition unambiguously; the latter, whether there are only temporal limits—not necessarily absolute—to the capacity of classical interpretations to communicate unambiguously that which we intended to capture within our formal expression.

We argued that, prima facie, applied science continues, perforce, to interpret mathematical concepts Platonically, whilst waiting for mathematics to provide suitable, and hopefully reliable, answers as to how best it may faithfully express its observations verifiably.

$\S 1.2$ Are axiomatic computational concepts really unambiguous?

This now raises the corresponding philosophical question that is implicit in Selmer Bringsjord’s narrational case against Church’s Thesis [1]:

Query 2 Is there a duality in the classical acceptance of non-constructive, foundational, concepts as axiomatic?

We now argue that beyond the question raised in an earlier post of whether—as computer scientist Lance Fortnow believes—Turing machines can capture everything we can compute’, or whether—as computer scientists Peter Wegner and Dina Goldin suggest—they are inappropriate as a universal foundation for computational problem solving’, we also need to address the philosophical question—implicit in Bringsjord’s paper—of whether, or not, the concept of effective computability’ is capable of a constructive, and intuitionistically unobjectionable, definition; and the relation of such definition to that of formal provability and to the standard perceptions of the Church and Turing Theses as reviewed here by at heart Trinity mathmo and by profession Philosopher Peter Smith.

We therefore consider the case for introduction of such a definition from a philosophical point of view, and consider some consequences.

$\S 1.2.1$ Mendelson’s thesis

We note that Elliott Mendelson [2] is quoted by Bringsjord in his paper as saying (italicised parenthetical qualifications added):

(i) “Here is the main conclusion I wish to draw:

it is completely unwarranted to say that CT is unprovable just because it states an equivalence between a vague, imprecise notion (effectively computable function) and a precise mathematical notion (partial-recursive function)”.

(ii) “The concepts and assumptions that support the notion of partial-recursive function are, in an essential way, no less vague and imprecise (non-constructive, and intuitionistically objectionable) than the notion of effectively computable function; the former are just more familiar and are part of a respectable theory with connections to other parts of logic and mathematics.

(The notion of effectively computable function could have been incorporated into an axiomatic presentation of classical mathematics, but the acceptance of CT made this unnecessary.) …

Functions are defined in terms of sets, but the concept of set is no clearer (not more non-constructive, and intuitionistically objectionable) than that of function and a foundation of mathematics can be based on a theory using function as primitive notion instead of set.

Tarski’s definition of truth is formulated in set-theoretic terms, but the notion of set is no clearer (not more non-constructive, and intuitionistically objectionable), than that of truth.

The model-theoretic definition of logical validity is based ultimately on set theory, the foundations of which are no clearer (not more non-constructive, and intuitionistically objectionable) than our intuitive (non-constructive, and intuitionistically objectionable) understanding of logical validity”.

(iii) “The notion of Turing-computable function is no clearer (not more non-constructive, and intuitionistically objectionable) than, nor more mathematically useful (foundationally speaking) than, the notion of an effectively computable function …”

where:

(a) The Church-Turing Thesis, CT, is formulated as:

“A function is effectively computable if and only if it is Turing-computable”.

(b) An effectively computable function is defined to be the computing of the function by an algorithm.

(c) The classical notion of an algorithm is expressed by Mendelson as:

“… an effective and completely specified procedure for solving a whole class of problems. …

An algorithm does not require ingenuity; its application is prescribed in advance and does not depend upon any empirical or random factors”.

and, where Bringsjord paraphrases (iii) as:

(iv) “The notion of a formally defined program for guiding the operation of a TM is no clearer than, nor more mathematically useful (foundationally speaking) than, the notion of an algorithm”.

adding that:

(v) “This proposition, it would then seem, is the very heart of the matter.

If (iv) is true then Mendelson has made his case; if this proposition is false, then his case is doomed, since we can chain back by modus tollens and negate (iii)”.

$\S 1.2.2$ The concept of constructive, and intuitionistically unobjectionable’

Now, prima facie, any formalisation of a vague and imprecise’, intuitive’ concept—say C—would normally be intended to capture the concept C both faithfully and completely within a constructive, and intuitionistically unobjectionable [3], language L.

Clearly, we could disprove the thesis—that C and its formalisation L are interchangeable, hence equivalent—by showing that there is a constructive aspect of C that is formalisable in a constructive language L’, but that such formalisation cannot be assumed expressible in L without introducing inconsistency.

However, equally clearly, there can be no way of proving the equivalence as this would contradict the premise that the concept is vague and imprecise’, hence essentially open-ended in a non-definable way, and so non-formalisable.

Obviously, Mendelson’s assertion that there is no justification for claiming Church’s Thesis as unprovable must, therefore, rely on an interpretation that differs significantly from the above; for instance, his concept of provability may appeal to the axiomatic acceptability of vague and imprecise’ concepts—as suggested by his remarks.

Now, we note that all the examples cited by Mendelson involve the decidability (computability) of an infinitude of meta-mathematical instances, where the distinction between the constructive meta-assertion that any given instance is individually decidable (instantiationally computable), and the non-constructive meta-assertion that all the instances are jointly decidable (uniformly computable), is not addressed explicitly.

However, $\S 1.2.1(a)$, $\S 1.2.1(b)$ and $\S 1.2.1(c)$ appear to suggest that Mendelson’s remarks relate implicitly to non-constructive meta-assertions.

Perhaps the real issue, then, is the one that emerges if we replace Mendelson’s use of implicitly open-ended concepts such as vague and imprecise’ and intuitive’ by the more meta-mathematically meaningful concept of non-constructive, and intuitionistically objectionable’, as italicised and indicated parenthetically.

The essence of Mendelson’s meta-assertion $\S 1.2.1(iii)$ then appears to be that, if the classically accepted definitions of foundational concepts such as partial recursive function’, function’, Tarskian truth’ etc. are also non-constructive, and intuitionistically objectionable, then replacing one non-constructive concept by another may be psychologically unappealing, but it should be meta-mathematically valid and acceptable.

$\S 1.2.3$ The duality

Clearly, meta-assertion $\S 1.2.1(iii)$ would stand refuted by a non-algorithmic’ effective method that is constructive.

However, if it is explicitly—and, as suggested by the nature of the arguments in Bringsjord’s paper, widely—accepted at the outset that any effective method is necessarily algorithmic (i.e. uniform as stated in $\S 1.2.1(c)$ above), then any counter-argument to CT can, prima facie, only offer non-algorithmic methods that may, paradoxically, be effective’ intuitively but in a non-constructive, and intuitionistically objectionable, way only!

Recognition of this dilemma is implicit in the admission that the various arguments, as presented by Bringsjord in the case against Church’s Thesis—including his narrational case—are open to reasonable, but inconclusive, refutations.

Nevertheless, if we accept Mendelson’s thesis that the inter-changeability of non-constructive concepts is valid in the foundations of mathematics, then Bringsjord’s case against Church’s Thesis, since it is based similarly on non-constructive concepts, should also be considered conclusive classically (even though it cannot, prima facie, be considered constructively conclusive in an intuitionistically unobjectionable way).

There is, thus, an apparent duality in the—seemingly extra-logical—decision as to whether an argument based on non-constructive concepts may be accepted as classically conclusive or not.

That this duality may originate in the very issues raised in Mendelson’s remarks—concerning the non-constructive roots of foundational concepts that are classically accepted as mathematically sound—is seen if we note that these issues may be more significant than is, prima facie, apparent.

$\S 1.2.4$ Definition of a formal mathematical object, and consequences

Thus, if we define a formal mathematical object as any symbol for an individual letter, function letter or a predicate letter that can be introduced through definition into a formal theory without inviting inconsistency, then it can be argued that unrestricted, non-constructive, definitions of non-constructive, foundational, set-theoretic concepts—such as mapping’, function’, recursively enumerable set’, etc.—in terms of constructive number-theoretic concepts—such as recursive number-theoretic functions and relations—may not always correspond to formal mathematical objects.

In other words, the assumption that every definition corresponds to a formal mathematical object may introduce a formal inconsistency into standard Peano Arithmetic and, ipso facto, into any Axiomatic Set Theory that models standard PA (an inconsistency which, loosely speaking, may be viewed as a constructive arithmetical parallel to Russell’s non-constructive impredicative set).

Since it can also be argued that the non-constructive element in Tarski’s definitions of satisfiability’ and truth’, and in Church’s Thesis, originate in a common, but removable, ambiguity in the interpretation of an effective method, perhaps it is worth considering whether Bringsjord’s acceptance of the assumption—that every constructive effective method is necessarily algorithmic, in the sense of being a uniform procedure as in $\S 1.2.1(c)$ above—is mathematically necessary, or even whether it is at all intuitively tenable.

(Uniform procedure: A property usually taken to be a necessary condition for a procedure to qualify as effective.)

Thus, we may argue that we can explicitly and constructively define a non-algorithmic’ effective method as one that, in any given instance, is instantiationally computable if, and only if, it terminates finitely with a conclusive result; and an algorithmic’ effective method as one that is uniformly computable if, and only if, it terminates finitely, with a conclusive result, in any given instance. [4]

$\S 1.2.5$ Bringsjord’s case against CT

Apropos the specific arguments against CT it would seem, prima facie, that a non-algorithmic effective—even if not obviously constructive—method could be implicit in the following argument considered by Bringsjord:

“Assume for the sake of argument that all human cognition consists in the execution of effective processes (in brains, perhaps). It would then follow by CT that such processes are Turing-computable, i.e., that computationalism is true. However, if computationalism is false, while there remains incontrovertible evidence that human cognition consists in the execution of effective processes, CT is overthrown”.

Assuming computationalism is false, the issue in this argument would, then, be whether there is a constructive, and adequate, expression of human cognition in terms of individually effective methods.

An appeal to such a non-algorithmic effective method may, in fact, be implicit in Bringsjord’s consideration of the predicate H, defined by:

$H(P, i)$ iff $(\exists n)S(P, i, n)$

where the predicate $S(P, u, n)$ holds if, and only if, TM M, running program P on input $u$, halts in exactly $n$ steps ($= MP : u =>n$ halt).

Bringsjord’s Total Computability

Bringsjord defines S as totally (and, implicitly, uniformly) computable in the sense that, given some triple $(P, u, n)$, there is some (uniform) program P* which, running on some TM M*, can infallibly give us a verdict, Y (yes’) or N (no’), for whether or not S is true of this triple.

He then notes that, since the ability to (uniformly) determine, for a pair $(P, i)$, whether or not H is true of it, is equivalent to solving the full halting problem, H is not totally computable.

Bringsjord’s Partial Computability

However, he also notes that there is a program (implicitly non-uniform, and so, possibly, effective individually) which, when asked whether or not some TM M run by P on $u$ halts, will produce Y iff $MP : u =>n$ halt. For this reason H is declared partially (implicitly, individually) computable.

More explicitly, Bringsjord remarks that Laszlo Kalmár’s refutation of CT [5] is classically inconclusive mainly because it does not admit any uniform effective method, but appeals to the existence of an infinitude of individually effective methods.

Kalmár’s Perspective of CT

Considering that it always seeks to calculate $g(n)$ (defined below) constructively even in the absence of a uniform procedure—not necessarily within a fixed postulate system—we shall show that Kalmár’s finitary argument ([6]as reproduced below from Bringsjord’s paper) makes a pertinent observation:

“First, he draws our attention to a function $g$ that isn’t Turing-computable, given that $f$ is [7]:

$g(x) = \mu y(f(x, y) = 0)$ = {the least $y$ such that $f(x, y) = 0$ if $y$ exists; and $0$ if there is no such $y$}

Kalmár proceeds to point out that for any $n$ in $N$ for which a natural number $y$ with $f(n, y) = 0$ exists,

an obvious method for the calculation of the least such $y$ … can be given,’

namely, calculate in succession the values $f(n, 0), f(n, 1), f(n, 2), \ldots$ (which, by hypothesis, is something a computist or TM can do) until we hit a natural number $m$ such that $f(n, m) = 0$, and set $y = m$.

On the other hand, for any natural number $n$ for which we can prove, not in the frame of some fixed postulate system but by means of arbitrary—of course, correct—arguments that no natural number $y$ with $f(n, y) = 0$ exists, we have also a method to calculate the value $g(n)$ in a finite number of steps.

Kalmár goes on to argue as follows. The definition of $g$ itself implies the tertium non datur, and from it and CT we can infer the existence of a natural number $p$ which is such that

(*) there is no natural number $y$ such that $f(p, y)= 0$; and

(**) this cannot be proved by any correct means.

Kalmár claims that (*) and (**) are very strange, and that therefore CT is at the very least implausible.”

Distinguishing between individually effective computability and uniformly effective computability

Now, the significant point that emerges from Bringsjord’s and Kalmár’s philosophical arguments is the need to distinguish formally between non-algorithmic (i.e., instantiational) computability (or, more precisely, algorithmic verifiability as defined here) and algorithmic (i.e., uniform) computability (or, more precisely, algorithmic computability as defined here) as highlighted in the Birmingham paper.

In the next post we note that this point has also been raised from a more formal, logical, perspective; and consider what is arguably an intuitively-more-adequate formal definability of effective computability’ in terms of ‘algorithmic verifiability’ under which the classical Church and Turing Theses do not hold, but weakened Church and Turing Theses do!

References

BBJ03 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic (4th ed). Cambridge University Press, Cambridge.

Bri93 Selmer Bringsjord. 1993. The Narrational Case Against Church’s Thesis. Easter APA meetings, Atlanta.

Ch36 Alonzo Church. 1936. An unsolvable problem of elementary number theory. In M. Davis (ed.). 1965. The Undecidable Raven Press, New York. Reprinted from the Am. J. Math., Vol. 58, pp.345-363.

Ct75 Gregory J. Chaitin. 1975. A Theory of Program Size Formally Identical to Information Theory. J. Assoc. Comput. Mach. 22 (1975), pp. 329-340.

Go31 Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York.

Go51 Kurt Gödel. 1951. Some basic theorems on the foundations of mathematics and their implications. Gibbs lecture. In Kurt Gödel, Collected Works III, pp.304-323.\ 1995. Unpublished Essays and Lectures. Solomon Feferman et al (ed.). Oxford University Press, New York.

Ka59 Laszlo Kalmár. 1959. An Argument Against the Plausibility of Church’s Thesis. In Heyting, A. (ed.) Constructivity in Mathematics. North-Holland, Amsterdam.

Kl36 Stephen Cole Kleene. 1936. General Recursive Functions of Natural Numbers. Math. Annalen vol. 112 (1936) pp.727-766.

Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton.

Me90 Elliott Mendelson. 1990. Second Thoughts About Church’s Thesis and Mathematical Proofs. Journal of Philosophy 87.5.

Pa71 Rohit Parikh. 1971. Existence and Feasibility in Arithmetic. The Journal of Symbolic Logic, Vol.36, No. 3 (Sep., 1971), pp. 494-508.

Si97 Wilfried Sieg. 1997. Step by recursive step: Church’s analysis of effective calculability Bulletin of Symbolic Logic, Volume 3, Number 2.

Sm07 Peter Smith. 2007. Church’s Thesis after 70 Years. A commentary and critical review of Church’s Thesis After 70 Years. In Meinong Studies Vol 1 (Ontos Mathematical Logic 1), 2006 (2013), Eds. Adam Olszewski, Jan Wolenski, Robert Janusz. Ontos Verlag (Walter de Gruyter), Frankfurt, Germany.

Tu36 Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

An12 … 2012. Evidence-Based Interpretations of PA. In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

Notes

Return to 2: Me90.

Return to 3: The terms constructive’ and constructive, and intuitionistically unobjectionable’ are used synonymously both in their familiar linguistic sense, and in a mathematically precise sense. Mathematically, we term a concept as constructive, and intuitionistically unobjectionable’ if, and only if, it can be defined in terms of pre-existing concepts without inviting inconsistency. Otherwise, we understand it to mean unambiguously verifiable, by some effective method’, within some finite, well-defined, language or meta-language. It may also be taken to correspond, broadly, to the concept of constructive, and intuitionistically unobjectionable’ in the sense apparently intended by Gödel in his seminal 1931 paper Go31, p.26.

Return to 4: We note that the possibility of a distinction between the interpreted number-theoretic meta-assertions, For any given natural number $x$, $F(x)$ is true’ and $F(x)$ is true for all natural numbers $x$‘, is not evident unless these are expressed symbolically as, $(\forall x)(F(x)$ is true)’ and $(\forall x)F(x)$ is true’, respectively. The issue, then, is whether the distinction can be given any mathematical significance. For instance, under a constructive formulation of Tarski’s definitions, we may qualify the latter by saying that it can be meaningfully asserted as a totality only if $F$‘ is a well-defined mathematical object.

Return to 5: Ka59.

Return to 6: Ka59.

Return to 7: Bringsjord notes that the original proof can be found on page 741 of Kleene Kl36.

Return to 8: We detail a formal proof of this Thesis in this post.

Author’s working archives & abstracts of investigations

So where exactly does the buck stop?

Another reason why Lucas and Penrose should not be faulted for continuing to believe in their well-known Gödelian arguments against computationalism lies in the lack of an adequate consensus on the concept of effective computability’.

For instance, Boolos, Burgess and Jeffrey (2003: Computability and Logic, 4th ed.~CUP, p37) define a diagonal halting function, $d$, any value of which can be computed effectively, although there is no single algorithm that can effectively compute $d$.

“According to Turing’s Thesis, since $d$ is not Turing-computable, $d$ cannot be effectively computable. Why not? After all, although no Turing machine computes the function $d$, we were able to compute at least its first few values, For since, as we have noted, $f_{1} = f_{2} = f_{3} =$ the empty function we have $d(1) = d(2) = d(3) = 1$. And it may seem that we can actually compute $d(n)$ for any positive integer $n$—if we don’t run out of time.”
… ibid. 2003. p37.

Now, the straightforward way of expressing this phenomenon should be to say that there are well-defined real numbers that are instantiationally computable, but not algorithmically computable.

Yet, following Church and Turing, such functions are labeled as effectively uncomputable!

The issue here seems to be that, when using language to express the abstract objects of our individual, and common, mental concept spaces’, we use the word exists’ loosely in three senses, without making explicit distinctions between them.

First, we may mean that an individually conceivable object exists, within a language $L$, if it lies within the range of the variables of $L$. The existence of such objects is necessarily derived from the grammar, and rules of construction, of the appropriate constant terms of the language—generally finitary in recursively defined languages—and can be termed as constructive in $L$ by definition.

Second, we may mean that an individually conceivable object exists, under a formal interpretation of $L$ in another formal language, say $L$, if it lies within the range of a variable of $L$ under the interpretation.

Again, the existence of such an object in $L$ is necessarily derivable from the grammar, and rules of construction, of the appropriate constant terms of $L$, and can be termed as constructive in $L$ by definition.

Third, we may mean that an individually conceivable object exists, in an interpretation $M$ of $L$, if it lies within the range of an interpreted variable of $L$, where $M$ is a Platonic interpretation of $L$ in an individual’s subjective mental conception (in Brouwer’s sense).

Clearly, the debatable issue is the third case.

So the question is whether we can—and, if so, how we may—correspond the Platonically conceivable objects of various individual interpretations of $L$, say $M$, $M$, $M$, …, unambiguously to the mathematical objects that are definable as the constant terms of $L$.

If we can achieve this, we can then attempt to relate $L$ to a common external world and try to communicate effectively about our individual mental concepts of the world that we accept as lying, by consensus, in a common, Platonic, concept-space’.

For mathematical languages, such a common concept-space’ is implicitly accepted as the collection of individual intuitive, Platonically conceivable, perceptions—$M$, $M$, $M$, …,—of the standard intuitive interpretation, say $M$, of Dedekind’s axiomatic formulation of the Peano Postulates.

Reasonably, if we intend a language or a set of languages to be adequate, first, for the expression of the abstract concepts of collective individual consciousnesses, and, second, for the unambiguous and effective communication of those of such concepts that we can accept as lying within our common concept-space, then we need to give effective guidelines for determining the Platonically conceivable mathematical objects of an individual perception of $M$ that we can agree upon, by common consensus, as corresponding to the constants (mathematical objects) definable within the language.

Now, in the case of mathematical languages in standard expositions of classical theory, this role is sought to be filled by the Church-Turing Thesis (CT). Its standard formulation postulates that every number-theoretic function (or relation, treated as a Boolean function) of $M$, which can intuitively be termed as effectively computable, is partial recursive / Turing-computable.

However, CT does not succeed in its objective completely.

Thus, even if we accept CT, we still cannot conclude that we have specified explicitly that the domain of $M$ consists of only constructive mathematical objects that can be represented in the most basic of our formal mathematical languages, namely, first-order Peano Arithmetic (PA) and Recursive Arithmetic (RA).

The reason seems to be that CT is postulated as a strong identity, which, prima facie, goes beyond the minimum requirements for the correspondence between the Platonically conceivable mathematical objects of $M$ and those of PA and RA.

“We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers.”
… Church 1936: An unsolvable problem of elementary number theory, Am.~J.~Math., Vol.~58, pp.~345–363.

“The theorem that all effectively calculable sequences are computable and its converse are proved below in outline.
… Turing 1936: On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, ser.~2.~vol.~42 (1936–7), pp.~230–265.

This violation of the principle of Occam’s Razor is highlighted if we note (e.g., Gödel 1931: On undecidable propositions of Principia Mathematica and related systems I, Theorem VII) that, pedantically, every recursive function (or relation) is not shown as identical to a unique arithmetical function (or relation), but (see the comment following Lemma 9 of this paper) only as instantiationally equivalent to an infinity of arithmetical functions (or relations).

Now, the standard form of CT only postulates algorithmically computable number-theoretic functions of $M$ as effectively computable.

It overlooks the possibility that there may be number-theoretic functions and relations which are effectively computable / decidable instantiationally in a Tarskian sense, but not algorithmically.

References

BBJ03 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic. (4th ed). Cambridge University Press, Cambridge.

Go31 Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. pp.5-38.

Lu61 John Randolph Lucas. 1961. Minds, Machines and Gödel. In Philosophy. Vol. 36, No. 137 (Apr. – Jul., 1961), pp. 112-127, Cambridge University Press.

Lu03 John Randolph Lucas. 2003. The Gödelian Argument: Turn Over the Page. In Etica & Politica / Ethics & Politics, 2003, 1.

Lu06 John Randolph Lucas. 2006. Reason and Reality. Edited by Charles Tandy. Ria University Press, Palo Alto, California.

Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand. pp.145-146.

Pe90 Roger Penrose. 1990. The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics. 1990, Vintage edition. Oxford University Press.

Pe94 Roger Penrose. 1994. Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press.

Sc67 Joseph R. Schoenfield. 1967. Mathematical Logic. Reprinted 2001. A. K. Peters Ltd., Massachusetts.

Ta33 Alfred Tarski. 1933. The concept of truth in the languages of the deductive sciences. In Logic, Semantics, Metamathematics, papers from 1923 to 1938. (p152-278). ed. John Corcoran. 1983. Hackett Publishing Company, Indianapolis.

Wa63 Hao Wang. 1963. A survey of Mathematical Logic. North Holland Publishing Company, Amsterdam.

An07a Bhupinder Singh Anand. 2007. The Mechanist’s challenge. In The Reasoner, Vol(1)5 p5-6.

An12 … 2012. Evidence-Based Interpretations of PA. In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

Author’s working archives & abstracts of investigations

$\S 1$ The Holy Grail of Arithmetic: Bridging Provability and Computability

See also this update.

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

Peter Wegner and Dina Goldin

In a short opinion paper, Computation Beyond Turing Machines‘, Computer Scientists Peter Wegner and Dina Goldin (Wg03) advanced the thesis that:

A paradigm shift is necessary in our notion of computational problem solving, so it can provide a complete model for the services of today’s computing systems and software agents.’

We note that Wegner and Goldin’s arguments, in support of their thesis, seem to reflect an extraordinarily eclectic view of mathematics, combining both an implicit acceptance of, and implicit frustration at, the standard interpretations and dogmas of classical mathematical theory:

(i) … Turing machines are inappropriate as a universal foundation for computational problem solving, and … computer science is a fundamentally non-mathematical discipline.’

(ii) (Turing’s) 1936 paper … proved that mathematics could not be completely modeled by computers.’

(iii) … the Church-Turing Thesis … equated logic, lambda calculus, Turing machines, and algorithmic computing as equivalent mechanisms of problem solving.’

(iv) Turing implied in his 1936 paper that Turing machines … could not provide a model for all forms of mathematics.’

(v) … Gödel had shown in 1931 that logic cannot model mathematics … and Turing showed that neither logic nor algorithms can completely model computing and human thought.’

These remarks vividly illustrate the dilemma with which not only Theoretical Computer Sciences, but all applied sciences that depend on mathematics—for providing a verifiable language to express their observations precisely—are faced:

Query: Are formal classical theories essentially unable to adequately express the extent and range of human cognition, or does the problem lie in the way formal theories are classically interpreted at the moment?

The former addresses the question of whether there are absolute limits on our capacity to express human cognition unambiguously; the latter, whether there are only temporal limits—not necessarily absolute—to the capacity of classical interpretations to communicate unambiguously that which we intended to capture within our formal expression.

Prima facie, applied science continues, perforce, to interpret mathematical concepts Platonically, whilst waiting for mathematics to provide suitable, and hopefully reliable, answers as to how best it may faithfully express its observations verifiably.

Lance Fortnow

This dilemma is also reflected in Computer Scientist Lance Fortnow’s on-line rebuttal of Wegner and Goldin’s thesis, and of their reasoning.

Thus Fortnow divides his faith between the standard interpretations of classical mathematics (and, possibly, the standard set-theoretical models of formal systems such as standard Peano Arithmetic), and the classical computational theory of Turing machines.

He relies on the former to provide all the proofs that matter:

Not every mathematical statement has a logical proof, but logic does capture everything we can prove in mathematics, which is really what matters’;

and, on the latter to take care of all essential, non-provable, truth:

… what we can compute is what computer science is all about’.

Can faith alone suffice?

However, as we shall argue in a subsequent post, Fortnow’s faith in a classical Church-Turing Thesis that ensures:

… Turing machines capture everything we can compute’,

may be as misplaced as his faith in the infallibility of standard interpretations of classical mathematics.

The reason: There are, prima facie, reasonably strong arguments for a Kuhnian (Ku62) paradigm shift; not, as Wegner and Goldin believe, in the notion of computational problem solving, but in the standard interpretations of classical mathematical concepts.

However, Wegner and Goldin could be right in arguing that the direction of such a shift must be towards the incorporation of non-algorithmic effective methods into classical mathematical theory (as detailed in the Birmingham paper); presuming, from the following remarks, that this is, indeed, what external interactions’ are assumed to provide beyond classical Turing-computability:

(vi) … that Turing machine models could completely describe all forms of computation … contradicted Turing’s assertion that Turing machines could only formalize algorithmic problem solving … and became a dogmatic principle of the theory of computation’.

(vii) … interaction between the program and the world (environment) that takes place during the computation plays a key role that cannot be replaced by any set of inputs determined prior to the computation’.

(viii) … a theory of concurrency and interaction requires a new conceptual framework, not just a refinement of what we find natural for sequential [algorithmic] computing’.

(ix) … the assumption that all of computation can be algorithmically specified is still widely accepted’.

A widespread notion of particular interest, which seems to be recurrently implicit in Wegner and Goldin’s assertions too, is that mathematics is a dispensable tool of science, rather than its indispensable mother tongue.

Elliott Mendelson

However, the roots of such beliefs may also lie in ambiguities, in the classical definitions of foundational elements, that allow the introduction of non-constructive—hence non-verifiable, non-computational, ambiguous, and essentially Platonic—elements into the standard interpretations of classical mathematics.

For instance, in a 1990 philosophical reflection, Elliott Mendelson’s following remarks (in Me90; reproduced from Selmer Bringsjord (Br93)), implicitly imply that classical definitions of various foundational elements can be argued as being either ambiguous, or non-constructive, or both:

Here is the main conclusion I wish to draw: it is completely unwarranted to say that CT is unprovable just because it states an equivalence between a vague, imprecise notion (effectively computable function) and a precise mathematical notion (partial-recursive function). … The concepts and assumptions that support the notion of partial-recursive function are, in an essential way, no less vague and imprecise than the notion of effectively computable function; the former are just more familiar and are part of a respectable theory with connections to other parts of logic and mathematics. (The notion of effectively computable function could have been incorporated into an axiomatic presentation of classical mathematics, but the acceptance of CT made this unnecessary.) … Functions are defined in terms of sets, but the concept of set is no clearer than that of function and a foundation of mathematics can be based on a theory using function as primitive notion instead of set. Tarski’s definition of truth is formulated in set-theoretic terms, but the notion of set is no clearer than that of truth. The model-theoretic definition of logical validity is based ultimately on set theory, the foundations of which are no clearer than our intuitive understanding of logical validity. … The notion of Turing-computable function is no clearer than, nor more mathematically useful (foundationally speaking) than, the notion of an effectively computable function.’

Consequently, standard interpretations of classical theory may, inadvertently, be weakening a desirable perception—of mathematics as the lingua franca of scientific expression—by ignoring the possibility that, since mathematics is, indeed, indisputably accepted as the language that most effectively expresses and communicates intuitive truth, the chasm between formal truth and provability must, of necessity, be bridgeable.

Cristian Calude, Elena Calude and Solomon Marcus

The belief in the existence of such a bridge is occasionally implicit in interpretations of computational theory.

For instance, in an arXived paper Passages of Proof, Computer Scientists Cristian Calude, Elena Calude and Solomon Marcus remark that:

“Classically, there are two equivalent ways to look at the mathematical notion of proof: logical, as a finite sequence of sentences strictly obeying some axioms and inference rules, and computational, as a specific type of computation. Indeed, from a proof given as a sequence of sentences one can easily construct a Turing machine producing that sequence as the result of some finite computation and, conversely, given a machine computing a proof we can just print all sentences produced during the computation and arrange them into a sequence.”

In other words, the authors seem to hold that Turing-computability of a proof’, in the case of an arithmetical proposition, is equivalent to provability of its representation in PA.

Wilfrid Sieg

We now attempt to build such a bridge formally, which is essentially one between the arithmetical ‘Decidability and Calculability’ described by Philosopher Wilfrid Sieg in his in-depth and wide-ranging survey ‘On Comptability‘, in which he addresses Gödel’s lifelong belief that an iff bridge between the two concepts is ‘impossible’ for ‘the whole calculus of predicates’ (Wi08, p.602).

$\S 2$ Bridging provability and computability: The foundations

In the paper titled “Evidence-Based Interpretations of $PA$” that was presented to the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, held from $2^{nd}$ to $6^{th}$ July 2012 at the University of Birmingham, UK (reproduced in this post) we have defined what it means for a number-theoretic function to be:

We have shown there that:

(i) The standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of the first order Peano Arithmetic PA is finitarily sound if, and only if, Aristotle’s particularisation holds over $N$; and the latter is the case if, and only if, PA is $\omega$-consistent.

(ii) We can define a finitarily sound algorithmic interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA over the domain $N$ where, if $[A]$ is an atomic formula $[A(x_{1}, x_{2}, \ldots, x_{n})]$ of PA, then the sequence of natural numbers $(a_{1}, a_{2}, \ldots, a_{n})$ satisfies $[A]$ if, and only if $[A(a_{1}, a_{2}, \ldots, a_{n})]$ is algorithmically computable under $\mathcal{I}_{PA(N,\ Algorithmic)}$, but we do not presume that Aristotle’s particularisation is valid over $N$.

(iii) The axioms of PA are always true under the finitary interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$, and the rules of inference of PA preserve the properties of satisfaction/truth under $\mathcal{I}_{PA(N,\ Algorithmic)}$.

We concluded that:

Theorem 1: The interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA is finitarily sound.

Theorem 2: PA is consistent.

$\S 3$ Extending Buss’ Bounded Arithmetic

One of the more significant consequences of the Birmingham paper is that we can extend the iff bridge between the domain of provability and that of computability envisaged under Buss’ Bounded Arithmetic by showing that an arithmetical formula $[F]$ is PA-provable if, and only if, $[F]$ interprets as true under an algorithmic interpretation of PA.

$\S 4$ A Provability Theorem for PA

We first show that PA can have no non-standard model (for a distinctly different proof of this convention-challenging thesis see this post and this paper), since it is algorithmically’ complete in the sense that:

Theorem 3: (Provability Theorem for PA) A PA formula $[F(x)]$ is PA-provable if, and only if, $[F(x)]$ is algorithmically computable as always true in $N$.

Proof: We have by definition that $[(\forall x)F(x)]$ interprets as true under the interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ if, and only if, $[F(x)]$ is algorithmically computable as always true in $N$.

Since $\mathcal{I}_{PA(N,\ Algorithmic)}$ is finitarily sound, it defines a finitary model of PA over $N$—say $\mathcal{M}_{PA(\beta)}$—such that:

If $[(\forall x)F(x)]$ is PA-provable, then $[F(x)]$ is algorithmically computable as always true in $N$;

If $[\neg(\forall x)F(x)]$ is PA-provable, then it is not the case that $[F(x)]$ is algorithmically computable as always true in $N$.

Now, we cannot have that both $[(\forall x)F(x)]$ and $[\neg(\forall x)F(x)]$ are PA-unprovable for some PA formula $[F(x)]$, as this would yield the contradiction:

(i) There is a finitary model—say $M1_{\beta}$—of PA+$[(\forall x)F(x)]$ in which $[F(x)]$ is algorithmically computable as always true in $N$.

(ii) There is a finitary model—say $M2_{\beta}$—of PA+$[\neg(\forall x)F(x)]$ in which it is not the case that $[F(x)]$ is algorithmically computable as always true in $N.$

The lemma follows. $\Box$

$\S 5$ The holy grail of arithmetic

We thus have that:

Corollary 1: PA is categorical finitarily.

Now we note that:

Lemma 2: If PA has a sound interpretation $\mathcal{I}_{PA(N,\ Sound)}$ over $N$, then there is a PA formula $[F]$ which is algorithmically verifiable as always true over $N$ under $\mathcal{I}_{PA(N,\ Sound)}$ even though $[F]$ is not PA-provable.

Proof In his seminal 1931 paper on formally undecidable arithmetical propositions, Kurt Gödel has shown how to construct an arithmetical formula with a single variable—say $[R(x)]$ [1]—such that $[R(x)]$ is not PA-provable [2], but $[R(n)]$ is instantiationally PA-provable for any given PA numeral $[n]$. Hence, for any given numeral $[n]$, the PA formula $xB \lceil [R(n)] \rceil$ must hold for some $x$. The lemma follows. $\Box$

By the argument in Theorem 3 it follows that:

Corollary 2: The PA formula $[\neg(\forall x)R(x)]$ defined in Lemma 2 is PA-provable.

Corollary 3: Under any sound interpretation of PA, Gödel’s $[R(x)]$ interprets as an algorithmically verifiable, but not algorithmically computable, tautology over $N$.

Proof Gödel has shown that $[R(x)]$ [3] interprets as an algorithmically verifiable tautology [4]. By Corollary 2 $[R(x)]$ is not algorithmically computable as always true in $N$. $\Box$

Corollary 4: PA is not $\omega$-consistent. [5]

Proof Gödel has shown that if PA is consistent, then $[R(n)]$ is PA-provable for any given PA numeral $[n]$ [6]. By Corollary 2 and the definition of $\omega$-consistency, if PA is consistent then it is not $\omega$-consistent. $\Box$

Corollary 5: The standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of PA is not finitarily sound, and does not yield a finitary model of PA [7].

Proof If PA is consistent but not $\omega$-consistent, then Aristotle’s particularisation does not hold over $N$. Since the standard’, interpretation of PA appeals to Aristotle’s particularisation, the lemma follows. $\Box$

Since formal quantification is currently interpreted in classical logic [8] so as to admit Aristotle’s particularisation over $N$ as axiomatic [9], the above suggests that we may need to review number-theoretic arguments [10] that appeal unrestrictedly to classical Aristotlean logic.

$\S 6$ The Provability Theorem for PA and Bounded Arithmetic

In a 1997 paper [11], Samuel R. Buss considered Bounded Arithmetics obtained by:

(a) limiting the applicability of the Induction Axiom Schema in PA only to functions with quantifiers bounded by an unspecified natural number bound $b$;

(b) weakening’ the statement of the axiom with the aim of differentiating between effective computability over the sequence of natural numbers, and feasible polynomial-time’ computability over a bounded sequence of the natural numbers [12].

Presumably Buss’ intent—as expressed below—is to build an iff bridge between provability in a Bounded Arithmetic and Computability so that a $\Pi_{k}$ formula, say $[(\forall x)f(x)]$, is provable in the Bounded Arithmetic if, and only if, there is an algorithm that, for any given numeral $[n]$, decides the $\Delta_{(k/(k-1))}$ formula $[f(n)]$ as true’:

If $[(\forall x)(\exists y)f(x, y)]$ is provable, then there should be an algorithm to find $y$ as a function of $x$ [13].

Since we have proven such a Provability Theorem for PA in the previous section, the first question arises:

$\S 7$ Does the introduction of bounded quantifiers yield any computational advantage?

Now, one difference [14] between a Bounded Arithmetic and PA is that we can presume in the Bounded Arithmetic that, from a proof of $[(\exists y)f(n, y)]$, we may always conclude that there is some numeral $[m]$ such that $[f(n, m)]$ is provable in the arithmetic; however, this is not a finitarily sound conclusion in PA.

Reason: Since $[(\exists y)f(n, y)]$ is simply a shorthand for $[\neg (\forall y)\neg f(n, y)]$, such a presumption implies that Aristotle’s particularisation holds over the natural numbers under any finitarily sound interpretation of PA.

To see that (as Brouwer steadfastly held) this may not always be the case, interpret $[(\forall x)f(x)]$ as [15]:

There is an algorithm that decides $[f(n)]$ as true’ for any given numeral $[n]$.

In such case, if $[(\forall x)(\exists y)f(x, y)]$ is provable in PA, then we can only conclude that:

There is an algorithm that, for any given numeral $[n]$, decides that it is not the case that there is an algorithm that, for any given numeral $[m]$, decides $[\neg f(n, m)]$ as true’.

We cannot, however, conclude—as we can in a Bounded Arithmetic—that:

There is an algorithm that, for any given numeral $[n]$, decides that there is an algorithm that, for some numeral $[m]$, decides $[f(n, m)]$ as true’.

Reason: $[(\exists y)f(n, y)]$ may be a Halting-type formula for some numeral $[n]$.

This could be the case if $[(\forall x)(\exists y)f(x, y)]$ were PA-unprovable, but $[(\exists y)f(n, y)]$ PA-provable for any given numeral $[n]$.

Presumably it is the belief that any finitarily sound interpretation of PA requires Aristotle’s particularisation to hold in $N$, and the recognition that the latter does not admit linking provability to computability in PA, which has led to considering the effect of bounding quantification in PA.

However, as we have seen in the preceding sections, we are able to link provability to computability through the Provability Theorem for PA by recognising precisely that, to the contrary, any interpretation of PA which requires Aristotle’s particularisation to hold in $N$ cannot be finitarily sound!

The postulation of an unspecified bound in a Bounded Arithmetic in order to arrive at a provability-computability link thus appears dispensible.

The question then arises:

$\S 8$ Does weakening’ the PA Induction Axiom Schema yield any computational advantage?

Now, Buss considers a bounded arithmetic $S_{2}$ which is, essentially, PA with the following weakened’ Induction Axiom Schema, PIND [16]:

$[\{f(0)\ \&\ (\forall x)(f(\lfloor \frac{x}{2} \rfloor) \rightarrow f(x))\} \rightarrow (\forall x)f(x)]$

However, PIND can be expressed in first-order Peano Arithmetic PA as follows:

$[\{f(0)\ \&\ (\forall x)(f(x) \rightarrow (f(2*x)\ \&\ f(2*x+1)))\} \rightarrow (\forall x)f(x)]$.

Moreover, the above is a particular case of PIND($k$):

$[\{f(0)\ \&\ (\forall x)(f(x) \rightarrow (f(k*x)\ \&\ f(k*x+1)\ \&\ \ldots\ \&\ f(k*x+k-1)))\}$ $\rightarrow (\forall x)f(x)]$.

Now we have the PA theorem:

$[(\forall x)f(x) \rightarrow \{f(0)\ \&\ (\forall x)(f(x) \rightarrow f(x+1))\}]$

It follows that the following is also a PA theorem:

$[\{f(0)\ \&\ (\forall x)(f(x) \rightarrow f(x+1))\} \rightarrow$ $\{f(0)\ \&\ (\forall x)(f(x) \rightarrow (f(k*x)\ \&\ f(k*x+1)\ \&\ \ldots\ \&\ f(k*x+k-1)))\}]$

In other words, for any numeral $[k]$, PIND($k$) is equivalent in PA to the standard Induction Axiom of PA!

Thus, the Provability Theorem for PA suggests that all arguments and conclusions of a Bounded Arithmetic can be reflected in PA without any loss of generality.

References

Br93 Selmer Bringsjord 1993. The Narrational Case Against Church’s Thesis. Easter APA meetings, Atlanta.

Br08 L. E. J. Brouwer. 1908. The Unreliability of the Logical Principles. English translation in A. Heyting, Ed. L. E. J. Brouwer: Collected Works 1: Philosophy and Foundations of Mathematics. Amsterdam: North Holland / New York: American Elsevier (1975): pp.107-111.

Bu97 Samuel R. Buss. 1997. Bounded Arithmetic and Propositional Proof Complexity. In Logic of Computation. pp.67-122. Ed. H. Schwichtenberg. Springer-Verlag, Berlin.

CCS01 Cristian S. Calude, Elena Calude and Solomon Marcus. 2001. Passages of Proof. Workshop, Annual Conference of the Australasian Association of Philosophy (New Zealand Division), Auckland. Archived at: http://arxiv.org/pdf/math/0305213.pdf. Also in EATCS Bulletin, Number 84, October 2004, viii+258 pp.

Da82 Martin Davis. 1958. Computability and Unsolvability. 1982 ed. Dover Publications, Inc., New York.

EC89 Richard L. Epstein, Walter A. Carnielli. 1989. Computability: Computable Functions, Logic, and the Foundations of Mathematics. Wadsworth & Brooks, California.

Go31 Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York.

HA28 David Hilbert & Wilhelm Ackermann. 1928. Principles of Mathematical Logic. Translation of the second edition of the Grundzüge Der Theoretischen Logik. 1928. Springer, Berlin. 1950. Chelsea Publishing Company, New York.

He04 Catherine Christer-Hennix. 2004. Some remarks on Finitistic Model Theory, Ultra-Intuitionism and the main problem of the Foundation of Mathematics. ILLC Seminar, 2nd April 2004, Amsterdam.

Hi25 David Hilbert. 1925. On the Infinite. Text of an address delivered in Münster on 4th June 1925 at a meeting of the Westphalian Mathematical Society. In Jean van Heijenoort. 1967.Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts.

Ku62 Thomas S. Kuhn. 1962. The structure of Scientific Revolutions. 2nd Ed. 1970. University of Chicago Press, Chicago.

Me90 Elliott Mendelson. 1990. Second Thoughts About Church’s Thesis and Mathematical Proofs. In Journal of Philosophy 87.5.

Pa71 Rohit Parikh. 1971. Existence and Feasibility in Arithmetic. In The Journal of Symbolic Logic,>i> Vol.36, No. 3 (Sep., 1971), pp. 494-508.

Rg87 Hartley Rogers Jr. 1987. Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, Massachusetts.

Ro36 J. Barkley Rosser. 1936. Extensions of some Theorems of Gödel and Church. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from The Journal of Symbolic Logic. Vol.1. pp.87-91.

Si08 Wilfrid Sieg. 2008. On Computability in Handbook of the Philosophy of Science. Philosophy of Mathematics. pp.525-621. Volume Editor: Andrew Irvine. General Editors: Dov M. Gabbay, Paul Thagard and John Woods. Elsevier BV. 2008.

WG03 Peter Wegner and Dina Goldin. 2003. Computation Beyond Turing Machines. Communications of the ACM, 46 (4) 2003.

Notes

Return to 1: Gödel refers to this formula only by its Gödel number $r$ (Go31, p.25(12)).

Return to 2: Gödel’s immediate aim in Go31 was to show that $[(\forall x)R(x)]$ is not P-provable; by Generalisation it follows, however, that $[R(x)]$ is also not P-provable.

Return to 3: Gödel refers to this formula only by its Gödel number $r$ (Go31, p.25, eqn.12).

Return to 4: Go31, p.26(2): “$(n)\neg(nB_{\kappa}(17Gen\ r))$ holds”.

Return to 5: This conclusion is contrary to accepted dogma. See, for instance, Davis’ remarks in Da82, p.129(iii) that:

“… there is no equivocation. Either an adequate arithmetical logic is $\omega$-inconsistent (in which case it is possible to prove false statements within it) or it has an unsolvable decision problem and is subject to the limitations of Gödel’s incompleteness theorem”.

Return to 6: Go31, p.26(2).

Return to 7: I note that finitists of all hues—ranging from Brouwer Br08 to Alexander Yessenin-Volpin He04—have persistently questioned the finitary soundness of the standard’ interpretation $\mathcal{I}_{PA(N,\ Standard)}$.

Return to 8: See Hi25, p.382; HA28, p.48; Be59, pp.178 \& 218.

Return to 9: In the sense of being intuitively obvious. See, for instance, Da82, p.xxiv; Rg87, p.308 (1)-(4); EC89, p.174 (4); BBJ03, p.102.

Return to 10: For instance Rosser’s construction of an undecidable arithmetical proposition in PA (see Ro36)—which does not explicitly assume that PA is $\omega$-consistent—implicitly presumes that Aristotle’s particularisation holds over $N$.

Return to 11: Bu97.

Return to 12: See also Pa71.

Return to 13: See Bu97.

Return to 14: We suspect the only one.

Return to 15: We have seen in the earlier sections that such an interpretation is finitarily sound.

Return to 16: Where $\lfloor \frac{x}{2} \rfloor$ denotes the largest natural number lower bound of the rational $\frac{x}{2}$.

Evidence-Based Interpretations of PA

In July 2012 I presented a paper titled “Evidence-Based Interpretations of $PA$” to the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, held from $2^{nd}$ to $6^{th}$ July 2012 at the University of Birmingham, UK.

The title was accurate pedantically, but deliberately misleading!

Misleading‘ because—had I kept the interest of the audience paramount—the more appropriate title should have been the provocative claim:

‘A solution to the Second of Hilbert’s Twenty Three Problems‘.

Deliberately‘ because whether or not the argumentation of the paper did lead to a finitary proof of consistency for $PA$ seemed, by itself, of little mathematical interest or consequence.

Reason: Because what did seem mathematically significant, however, was a distinction upon which the argumentation rested—between the use of algorithmic computability and algorithmic verifiabilty for logical validity—which had hitherto remained implicit.

Why the deception? Well, partly because caution was understandably advised, but to a larger extent because we know that either $PA$ has a sound interpretation, or it is inconsistent.

Now if—following David Hilbert’s line of reasoning in the Second of his celebrated Twenty Three problems—we embrace the former (since the latter seems u-u-u-un-un-un-unthinkable‘), then we must believe either that assignment of unique truth values to the $PA$ formulas under any sound interpretation of $PA$ is essentially human-intelligence subjective (eerily akin to a human revelation), or that there must be an objective such assignment that does not depend upon some unique way in which a human intelligence perceives and reasons.

The point of the conference paper was to highlight that the standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of $PA$ does not address this issue, since it is silent on the methodology for such an assignment.

$\mathcal{I}_{PA(N,\ Standard)}$ essentially asserts that, under Tarski’s inductive definitions of the satisfaction and truth of the formulas of $PA$ under the interpretation, if there is a methodology for uniquely defining the satisfaction and truth of the atomic formulas of $PA$ (presumably in a way that can be taken to mirror our—i.e. humankind’s—intuitive notion of the truth of the corresponding arithmetical propositions), then the satisfaction and truth of the compound PA formulas are defined uniquely under the interpretation by induction (and may also be taken to mirror our intuitive notion of the truth of the corresponding arithmetical propositions).

The consequences of not specifying a methodology are actually best illustrated by this example (in the borrowed terminology of a correspondent):

Let $E(A)$ mean that there is an assignment which provides objective evidence (o.e.) for $A$. It seems $\mathcal{I}_{PA(N,\ Standard)}$ breaks down in the general treatment of conditionals $A \rightarrow B$. In order to have $E(A \rightarrow B)$, we need to have an assignment which, first of all decides whether $E(A)$ and, if it verifies that, then verifies $E(B)$. But to decide whether $E(A)$ we have to decide whether or not there is an assignment that provides o.e. for $A$, and that can’t be done in general under $\mathcal{I}_{PA(N,\ Standard)}$ in the absence of a methodology for assigning objective satisfaction and truth values to the atomic formulas of $PA$.

The aim of the Birmingham paper (reproduced below) was to bridge this gap.

We formally showed there that there are, indeed, two essentially different methods of constructively assigning objective truth values uniquely to the atomic formulas of $PA$.

We then argued that if Tarski’s definitions are further accepted as inductively determining unique satisfaction and truth values for the compound formulas of $PA$, then we arrive at two interpretations of $PA$, say $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ and $\mathcal{I}_{PA(N,\ Algorithmic)}$.

We showed that $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ is sound if, and only if, $\mathcal{I}_{PA(N,\ Standard)}$ is sound; whence the latter, if sound, can indeed be taken to mirror our intuitive notion of the truth of the corresponding arithmetical propositions over the structure of the natural numbers as intended.

However, this interpretation is not finitary since the Axiom Schema of Finite Induction is not justified finitarily under the interpretation.

We further showed that $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ is sound if, and only if, $PA$ is $\omega$-consistent; which illuminates Gödel’s undecidability Theorem (if $PA$ is $\omega$-consistent, it must have a formally undecidable proposition).

We then argued that the other interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ is sound since the Axiom Schema of Finite Induction is justified finitarily under the interpretation.

However, what interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ can be taken to mirror is only the algorithmic decidability of our intuitive notion of the truth of the corresponding arithmetical propositions!

In other words, arithmetical provability can be taken to correspond not to our intuitive notion of arithmetical truth (which is subjective), but to the algorithmic decidability of our intuitive notion of arithmetical truth (which is objective).

Abstract

We shall now show formally that Tarski’s inductive definitions admit evidence-based interpretations of the first-order Peano Arithmetic PA that allow us to define the satisfaction and truth of the quantified formulas of PA constructively over the domain $N$ of the natural numbers in two essentially different ways:

(1) in terms of algorithmic verifiabilty; and

(2) in terms of algorithmic computability.

We shall argue that the algorithmically computable PA-formulas can provide a finitary interpretation of PA over the domain $N$ of the natural numbers from which we may conclude that PA is consistent.

1 Introduction

In this paper we seek to address one of the philosophical challenges associated with accepting arithmetical propositions as true under an interpretation—either axiomatically or on the basis of subjective self-evidence—without any effective methodology for objectively evidencing such acceptance [1].

For instance, conventional wisdom accepts Alfred Tarski’s definitions of the satisfiability and truth of the formulas of a formal language under an interpretation [2] and postulates that, under the standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of the first-order Peano Arithmetic PA [3] over the domain $N$ of the natural numbers:

(i) The atomic formulas of PA can be assumed as decidable under $\mathcal{I}_{PA(N,\ Standard)}$;

(ii) The PA axioms can be assumed to interpret as satisfied/true under $\mathcal{I}_{PA(N,\ Standard)}$;

(iii) the PA rules of inference—Generalisation and Modus Ponens—can be assumed to preserve such satisfaction/truth under $\mathcal{I}_{PA(N,\ Standard)}$.

Standard interpretation of PA: The standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of PA over the domain $N$ of the natural numbers is the one in which the logical constants have their usual’ interpretations [4] in Aristotle’s logic of predicates (which subsumes Aristotle’s particularisation [5]), and [7]:

(a) the set of non-negative integers is the domain;

(b) the symbol [0] interprets as the integer 0;

(c) the symbol $[']$ interprets as the successor operation (addition of 1);

(d) the symbols $[+]$ and $[\star]$ interpret as ordinary addition and multiplication;

(e) the symbol $[=]$ interprets as the identity relation.

The axioms of first-order Peano Arithmetic (PA)

PA$_{1}$ $[(x_{1} = x_{2}) \rightarrow ((x_{1} = x_{3}) \rightarrow (x_{2} = x_{3}))]$;

PA$_{2}$ $[(x_{1} = x_{2}) \rightarrow (x_{1}^{\prime} = x_{2}^{\prime})]$;

PA$_{3}$ $[0 \neq x_{1}^{\prime}]$;

PA$_{4}$ $[(x_{1}^{\prime} = x_{2}^{\prime}) \rightarrow (x_{1} = x_{2})]$;

PA$_{5}$ $[( x_{1} + 0) = x_{1}]$;

PA$_{6}$ $[(x_{1} + x_{2}^{\prime}) = (x_{1} + x_{2})^{\prime}]$;

PA$_{7}$ $[( x_{1} \star 0) = 0]$;

PA$_{8}$ $[( x_{1} \star x_{2}^{\prime}) = ((x_{1} \star x_{2}) + x_{1})]$;

PA$_{9}$ For any well-formed formula $[F(x)]$ of PA:

$[F(0) \rightarrow (((\forall x)(F(x) \rightarrow F(x^{\prime}))) \rightarrow (\forall x)F(x))]$.

Generalisation in PA: If $[A]$ is PA-provable, then so is $[(\forall x)A]$.

Modus Ponens in PA: If $[A]$ and $[A \rightarrow B]$ are PA-provable, then so is $[B]$.

We shall show that although the seemingly innocent and self-evident assumption in (i) can, indeed, be justified, it conceals an ambiguity whose impact on (ii) and (iii) is far-reaching in significance and needs to be made explicit.

Reason: Tarski’s inductive definitions admit evidence-based interpretations of PA that actually allow us to metamathematically define the satisfaction and truth of the atomic (and, ipso facto, quantified) formulas of PA constructively over $N$ in two essentially different ways as below, only one of which is finitary [7]:

(1) in terms of algorithmic verifiabilty [8];

(2) in terms of algorithmic computability [9].

Case 1: We show in Section 4.2 that the algorithmically verifiable PA-formulas admit an unusual, instantiational’ Tarskian interpretation $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ of PA over the domain $\mathbb {N}$ of the PA numerals; and that this interpretation is sound if, and only if, PA is $\omega$-consistent.

Soundness (formal system): We define a formal system S as sound under a Tarskian interpretation $\mathcal{I}_{S}$ over a domain $D$ if, and only if, every theorem $[T]$ of S translates as $[T]$ is true under $\mathcal{I}_{S}$ in $D$‘.

Soundness (interpretation): We define a Tarskian interpretation $\mathcal{I}_{S}$ of a formal system S as sound over a domain $D$ if, and only if, S is sound under the interpretation $\mathcal{\mathcal{I}_{S}}$ over the domain $D$.

Simple consistency: A formal system S is simply consistent if, and only if, there is no S-formula $[F(x)]$ for which both $[(\forall x)F(x)]$ and $[\neg(\forall x)F(x)]$ are S-provable.

$\omega$-consistency: A formal system S is $\omega$-consistent if, and only if, there is no S-formula $[F(x)]$ for which, first, $[\neg(\forall x)F(x)]$ is S-provable and, second, $[F(a)]$ is S-provable for any given S-term $[a]$.

We further show that this interpretation can be viewed as a formalisation of the standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of PA over $N$; in the sense that—under Tarski’s definitions—$\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ is sound over $\mathbb {N}$ if, and only if, $\mathcal{I}_{PA(N,\ Standard)}$ is sound over $N$ (as postulated in (ii) and (iii) above).

Although the standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ is assumed to be sound over $N$ (as expressed by (ii) and (iii) above), it cannot claim to be finitary since it it is not known to lead to a finitary justification of the truth—under Tarski’s definitions—of the Axiom Schema of (finite) Induction of PA in $N$ from which we may conclude—in an intuitionistically unobjectionable manner—that PA is consistent [10].

We note that Gerhard Gentzen’s constructive’ [11] consistency proof for formal number theory [12] is debatably finitary [13], since it involves a Rule of Infinite Induction that appeals to the properties of transfinite ordinals.

Case 2: We show further in Section 4.3 that the algorithmically computable PA-formulas admit an algorithmic’ Tarskian interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA over $N$.

We then argue in Section 5 that $\mathcal{I}_{PA(N,\ Algorithmic)}$ is essentially different from $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ since the PA-axioms—including the Axiom Schema of (finite) Induction—are algorithmically computable as satisfied/true under the standard interpretation of PA over $N$, and the PA rules of inference preserve algorithmically computable satisfiability/truth under the interpretation [14].

We conclude from the above that the interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ is finitary, and hence sound over $N$ [15].

We further conclude from the soundness of the interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ over $N$ that PA is consistent [16].

2 Interpretation of an arithmetical language in terms of the computations of a simple functional language

We begin by noting that we can, in principle, define [17] the classical satisfaction’ and truth’ of the formulas of a first order arithmetical language, such as PA, verifiably under an interpretation using as evidence [18] the computations of a simple functional language.

Such definitions follow straightforwardly for the atomic formulas of the language (i.e., those without the logical constants that correspond to negation’, conjunction’, implication’ and quantification’) from the standard definition of a simple functional language [19].

Moreover, it follows from Alfred Tarski’s seminal 1933 paper on the concept of truth in the languages of the deductive sciences [20] that the satisfaction’ and truth’ of those formulas of a first-order language which contain logical constants can be inductively defined, under an interpretation, in terms of the satisfaction’ and truth’ of the interpretations of only the atomic formulas of the language.

Hence the satisfaction’ and truth’ of those formulas (of an arithmetical language) which contain logical constants can, in principle, also be defined verifiably under an interpretation using as evidence the computations of a simple functional language.

We show in Section 4 that this is indeed the case for PA under its standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$, when this is explicitly defined as in Section 5.

We show, moreover, that we can further define algorithmic truth’ and algorithmic falsehood’ under $\mathcal{I}_{PA(N,\ Standard)}$ such that the PA axioms interpret as always algorithmically true, and the rules of inference preserve algorithmic truth, over the domain $N$ of the natural numbers.

2.1 The definitions of algorithmic truth’ and algorithmic falsehood’ under $\mathcal{I}_{PA(N,\ Standard)}$ are not symmetric with respect to truth’ and falsehood’ under $\mathcal{I}_{PA(N,\ Standard)}$

However, the definitions of algorithmic truth’ and algorithmic falsehood’ under $\mathcal{I}_{PA(N,\ Standard)}$ are not symmetric with respect to classical (verifiable) truth’ and falsehood’ under $\mathcal{I}_{PA(N,\ Standard)}$.

For instance, if a formula $[(\forall x)F(x)]$ of an arithmetic is algorithmically true under an interpretation (such as $\mathcal{I}_{PA(N,\ Standard)}$), then we may conclude that there is an algorithm that, for any given numeral $[a]$, provides evidence that the formula $[F(a)]$ is algorithmically true under the interpretation.

In other words, there is an algorithm that provides evidence that the interpretation $F^{*}(a)$ of $[F(a)]$ holds in $N$ for any given natural number $a$.

Notation: We use enclosing square brackets as in $[F(x)]$‘ to indicate that the expression inside the brackets is to be treated as denoting a formal expression (formal string) of a formal language. We use an asterisk as in $F^{*}(x)$‘ to indicate the asterisked expression $F^{*}(x)$ is to be treated as denoting the interpretation of the formula $[F(x)]$ in the corresponding domain of the interpretation.

Defining the term hold’: We define the term hold’—when used in connection with an interpretation of a formal language L and, more specifically, with reference to the computations of a simple functional language associated with the atomic formulas of the language L—explicitly in Section 4; the aim being to avoid appealing to the classically subjective (and existential) connotation implicitly associated with the term under an implicitly defined standard interpretation of an arithmetic [21].

However, if a formula $[(\forall x)F(x)]$ of an arithmetic is algorithmically false under an interpretation, then we can only conclude that there is no algorithm that, for any given natural number $a$, can provide evidence whether the interpretation $F^{*}(a)$ holds or not in $N$ .

We cannot conclude that there is a numeral $[a]$ such that the formula $[F(a)]$ is algorithmically false under the interpretation; nor can we conclude that there is a natural number $b$ such that $F^{*}(b)$ does not hold in $N$.

Such a conclusion would require:

(i) either some additional evidence that will verify for some assignment of numerical values to the free variables of $[F]$ that the corresponding interpretation $F^{*}$ does not hold [22];

(ii) or the additional assumption that either Aristotle’s particularisation holds over the domain of the interpretation (as is implicitly presumed under the standard interpretation of PA over $N$) or, equivalently, that the arithmetic is $\omega$-consistent [23].

Aristotle’s particularisation: This holds that from a meta-assertion such as:

It is not the case that: For any given $x$, $P^{*}(x)$ does not hold’,

usually denoted symbolically by $\neg(\forall x)\neg P^{*}(x)$‘, we may always validly infer in the classical, Aristotlean, logic of predicates [24] that:

There exists an unspecified $x$ such that $P^{*}(x)$ holds’,

usually denoted symbolically by $(\exists x)P^{*}(x)$‘.

The significance of Aristotle’s particularisation for the first-order predicate calculus: We note that in a formal language the formula $[(\exists x)P(x)]$‘ is an abbreviation for the formula $[\neg(\forall x)\neg P(x)]$‘. The commonly accepted interpretation of this formula—and a fundamental tenet of classical logic unrestrictedly adopted as intuitively obvious by standard literature [25] that seeks to build upon the formal first-order predicate calculus—tacitly appeals to Aristotlean particularisation.

However, L. E. J. Brouwer had noted in his seminal 1908 paper on the unreliability of logical principles [26] that the commonly accepted interpretation of this formula is ambiguous if interpretation is intended over an infinite domain.

Brouwer essentially argued that, even supposing the formula $[P(x)]$‘ of a formal Arithmetical language interprets as an arithmetical relation denoted by $P^{*}(x)$‘, and the formula $[\neg(\forall x)\neg P(x)]$‘ as the arithmetical proposition denoted by $\neg(\forall x)\neg P^{*}(x)$‘, the formula $[(\exists x)P(x)]$‘ need not interpret as the arithmetical proposition denoted by the usual abbreviation $(\exists x)P^{*}(x)$‘; and that such postulation is invalid as a general logical principle in the absence of a means for constructing some putative object $a$ for which the proposition $P^{*}(a)$ holds in the domain of the interpretation.

Hence we shall follow the convention that the assumption that $(\exists x)P^{*}(x)$‘ is the intended interpretation of the formula $[(\exists x)P(x)]$‘—which is essentially the assumption that Aristotle’s particularisation holds over the domain of the interpretation—must always be explicit.

The significance of Aristotle’s particularisation for PA: In order to avoid intuitionistic objections to his reasoning, Kurt Gödel introduced the syntactic property of $\omega$-consistency as an explicit assumption in his formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions [27].

Gödel explained at some length [28] that his reasons for introducing $\omega$-consistency explicitly was to avoid appealing to the semantic concept of classical arithmetical truth in Aristotle’s logic of predicates (which presumes Aristotle’s particularisation).

It is straightforward to show that the two concepts are meta-mathematically equivalent in the sense that, if PA is consistent, then PA is $\omega$-consistent if, and only if, Aristotle’s particularisation holds under the standard interpretation of PA over $N$.

3 Defining algorithmic verifiability and algorithmic computability

The asymmetry of Section 2.1 suggests the following two concepts [29]:

Definition 1: Algorithmic verifiability: An arithmetical formula $[(\forall x)F(x)]$ is algorithmically verifiable as true under an interpretation if, and only if, for any given numeral $[a]$, we can define an algorithm which provides evidence that $[F(a)]$ interprets as true under the interpretation.

Tarskian interpretation of an arithmetical language verifiably in terms of the computations of a simple functional language: We show in Section 4 that the algorithmic verifiability’ of the formulas of a formal language which contain logical constants can be inductively defined under an interpretation in terms of the algorithmic verifiability’ of the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under the standard interpretation of PA over $N$ if, and only if, they are algorithmically verifiable under the interpretation (Corollary 2).

Definition 2: Algorithmic computability: An arithmetical formula $[(\forall x)F(x)]$ is algorithmically computable as true under an interpretation if, and only if, we can define an algorithm that, for any given numeral $[a]$, provides evidence that $[F(a)]$ interprets as true under the interpretation.

Tarskian interpretation of an arithmetical language algorithmically in terms of the computations of a simple functional language: We show in Section 4 that the algorithmic computability’ of the formulas of a formal language which contain logical constants can also be inductively defined under an interpretation in terms of the algorithmic computability’ of the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under an algorithmic interpretation of PA over $N$ if, and only if, they are algorithmically computable under the interpretation .

We now show that the above concepts are well-defined under the standard interpretation of PA over $N$.

4 The implicit Satisfaction condition in Tarski’s inductive assignment of truth-values under an interpretation

We first consider the significance of the implicit Satisfaction condition in Tarski’s inductive assignment of truth-values under an interpretation.

We note that—essentially following standard expositions [30] of Tarski’s inductive definitions on the satisfiability’ and truth’ of the formulas of a formal language under an interpretation—we can define:

Definition 3: If $[A]$ is an atomic formula $[A(x_{1}, x_{2}, \ldots, x_{n})]$ of a formal language S, then the denumerable sequence $(a_{1}, a_{2}, \ldots)$ in the domain $D$ of an interpretation $\mathcal{I}_{S(D)}$ of S satisfies $[A]$ if, and only if:

(i) $[A(x_{1}, x_{2}, \ldots, x_{n})]$ interprets under $\mathcal{I}_{S(D)}$ as a unique relation $A^{*}(x_{1}, x_{2},$ $\ldots, x_{n})$ in $D$ for any witness $\mathcal{W}_{D}$ of $D$;

(ii) there is a Satisfaction Method, SM($\mathcal{I}_{S(D)}$) that provides objective evidence [30] by which any witness $\mathcal{W}_{D}$ of $D$ can objectively define for any atomic formula $[A(x_{1}, x_{2}, \ldots, x_{n})]$ of S, and any given denumerable sequence $(b_{1}, b_{2}, \ldots)$ of $D$, whether the proposition $A^{*}(b_{1}, b_{2}, \ldots, b_{n})$ holds or not in $D$;

(iii) $A^{*}(a_{1}, a_{2}, \ldots, a_{n})$ holds in $D$ for any $\mathcal{W}_{D}$.

Witness: From a constructive perspective, the existence of a witness’ as in (i) above is implicit in the usual expositions of Tarski’s definitions.

Satisfaction Method: From a constructive perspective, the existence of a Satisfaction Method as in (ii) above is also implicit in the usual expositions of Tarski’s definitions.

A constructive perspective: We highlight the word define‘ in (ii) above to emphasise the constructive perspective underlying this paper; which is that the concepts of satisfaction’ and truth’ under an interpretation are to be explicitly viewed as objective assignments by a convention that is witness-independent. A Platonist perspective would substitute decide’ for define’, thus implicitly suggesting that these concepts can exist’, in the sense of needing to be discovered by some witness-dependent means—eerily akin to a revelation’—if the domain $D$ is $N$.

We can now inductively assign truth values of satisfaction’, truth’, and falsity’ to the compound formulas of a first-order theory S under the interpretation $\mathcal{I}_{S(D)}$ in terms of only the satisfiability of the atomic formulas of S over $D$ as usual [31]:

Definition 4: A denumerable sequence $s$ of $D$ satisfies $[\neg A]$ under $\mathcal{I}_{S(D)}$ if, and only if, $s$ does not satisfy $[A]$;

Definition 5: A denumerable sequence $s$ of $D$ satisfies $[A \rightarrow B]$ under $\mathcal{I}_{S(D)}$ if, and only if, either it is not the case that $s$ satisfies $[A]$, or $s$ satisfies $[B]$;

Definition 6: A denumerable sequence $s$ of $D$ satisfies $[(\forall x_{i})A]$ under $\mathcal{I}_{S(D)}$ if, and only if, given any denumerable sequence $t$ of $D$ which differs from $s$ in at most the $i$‘th component, $t$ satisfies $[A]$;

Definition 7: A well-formed formula $[A]$ of $D$ is true under $\mathcal{I}_{S(D)}$ if, and only if, given any denumerable sequence $t$ of $D$, $t$ satisfies $[A]$;

Definition 8: A well-formed formula $[A]$ of $D$ is false under $\mathcal{I}_{S(D)}$ if, and only if, it is not the case that $[A]$ is true under $\mathcal{I}_{S(D)}$.

It follows that [32]:

Theorem 1: (Satisfaction Theorem) If, for any interpretation $\mathcal{I}_{S(D)}$ of a first-order theory S, there is a Satisfaction Method SM($\mathcal{I}_{S(D)}$) which holds for a witness $\mathcal{W}_{D}$ of $D$, then:

(i) The $\Delta_{0}$ formulas of S are decidable as either true or false over $D$ under $\mathcal{I}_{S(D)}$;

(ii) If the $\Delta_{n}$ formulas of S are decidable as either true or as false over $D$ under $\mathcal{I}_{S(D)}$, then so are the $\Delta(n+1)$ formulas of S.

Proof: It follows from the above definitions that:

(a) If, for any given atomic formula $[A(x_{1}, x_{2}, \ldots, x_{n})]$ of S, it is decidable by $\mathcal{W}_{D}$ whether or not a given denumerable sequence $(a_{1}, a_{2}, \ldots)$ of $D$ satisfies $[A(x_{1}, x_{2}, \ldots, x_{n})]$ in $D$ under $\mathcal{I}_{S(D)}$ then, for any given compound formula $[A^{1}(x_{1}, x_{2}, \ldots, x_{n})]$ of S containing any one of the logical constants $\neg, \rightarrow, \forall$, it is decidable by $\mathcal{W}_{D}$ whether or not $(a_{1}, a_{2}, \ldots)$ satisfies $[A^{1}(x_{1}, x_{2}, \ldots, x_{n})]$ in $D$ under $\mathcal{I}_{S(D)}$;

(b) If, for any given compound formula $[B^{n}(x_{1}, x_{2}, \ldots, x_{n})]$ of S containing $n$ of the logical constants $\neg, \rightarrow, \forall$, it is decidable by $\mathcal{W}_{D}$ whether or not a given denumerable sequence $(a_{1}, a_{2}, \ldots)$ of $D$ satisfies $[B^{n}(x_{1}, x_{2}, \ldots, x_{n})]$ in $D$ under $\mathcal{I}_{S(D)}$ then, for any given compound formula $[B^{(n+1)}(x_{1}, x_{2}, \ldots, x_{n})]$ of S containing $n+1$ of the logical constants $\neg, \rightarrow, \forall$, it is decidable by $\mathcal{W}_{D}$ whether or not $(a_{1}, a_{2}, \ldots)$ satisfies $[B^{(n+1)}(x_{1}, x_{2}, \ldots, x_{n})]$ in $D$ under $\mathcal{I}_{S(D)}$;

We thus have that:

(c) The $\Delta_{0}$ formulas of S are decidable by $\mathcal{W}_{D}$ as either true or false over $D$ under $\mathcal{I}_{S(D)}$;

(d) If the $\Delta_{n}$ formulas of S are decidable by $\mathcal{W}_{D}$ as either true or as false over $D$ under $\mathcal{I}_{S(D)}$, then so are the $\Delta(n+1)$ formulas of S. $\Box$

In other words, if the atomic formulas of of S interpret under $\mathcal{I}_{S(D)}$ as decidable with respect to the Satisfaction Method SM($\mathcal{I}_{S(D)}$) by a witness $\mathcal{W}_{D}$ over some domain $D$, then the propositions of S (i.e., the $\Pi_{n}$ and $\Sigma_{n}$ formulas of S) also interpret as decidable with respect to SM($\mathcal{I}_{S(D)}$) by the witness $\mathcal{W}_{D}$ over $D$.

We now consider the application of Tarski’s definitions to various interpretations of first-order Peano Arithmetic PA.

4.1 The standard interpretation of PA over the domain $N$ of the natural numbers

The standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of PA over the domain $N$ of the natural numbers is obtained if, in $\mathcal{I}_{S(D)}$:

(a) we define S as PA with standard first-order predicate calculus as the underlying logic [34];

(b) we define $D$ as the set $N$ of natural numbers;

(c) for any atomic formula $[A(x_{1}, x_{2}, \ldots, x_{n})]$ of PA and sequence $(a_{1}, a_{2}, \ldots, a_{n})$ of $N$, we take $\|$SATCON($\mathcal{I}_{PA(N)}$)$\|$ as:

$\|$$A^{*}(a_{1}^{*}, a_{2}^{*}, \ldots, a_{n}^{*})$ holds in $N$ and, for any given sequence $(b_{1}^{*}, b_{2}^{*}, \ldots, b_{n}^{*})$ of $N$, the proposition $A^{*}(b_{1}^{*}, b_{2}^{*}, \ldots, b_{n}^{*})$ is decidable in $N$$\|$;

(d) we define the witness $\mathcal{W}_{(N,\ Standard)}$ informally as the mathematical intuition’ of a human intelligence for whom, classically, $\|$SATCON($\mathcal{I}_{PA(N)}$)$\|$ has been implicitly accepted as objectively decidable’ in $N$;

We shall show that such acceptance is justified, but needs to be made explicit since:

Lemma 1: $A^{*}(x_{1}, x_{2}, \ldots, x_{n})$ is both algorithmically verifiable and algorithmically computable in $N$ by $\mathcal{W}_{(N,\ Standard)}$.

Proof: (i) It follows from the argument in Theorem 2 (below) that $A^{*}(x_{1}, x_{2}, \ldots, x_{n})$ is algorithmically verifiable in $N$ by $\mathcal{W}_{(N,\ Standard)}$.

(ii) It follows from the argument in Theorem 3 (below) that $A^{*}(x_{1}, x_{2},$ $\ldots, x_{n})$ is algorithmically computable in $N$ by $\mathcal{W}_{(N,\ Standard)}$. The lemma follows. $\Box$

Now, although it is not immediately obvious from the standard interpretation of PA over $N$ which of (i) or (ii) may be taken for explicitly deciding $\|$SATCON($\mathcal{I}_{PA(N)}$)$\|$ by the witness $\mathcal{W}_{(N,\ Standard)}$, we shall show in Section 4.2 that (i) is consistent with (e) below; and in Section 4.3 that (ii) is inconsistent with (e). Thus the standard interpretation of PA over $N$ implicitly presumes (i).

(e) we postulate that Aristotle’s particularisation holds over $N$ [35].

Clearly, (e) does not form any part of Tarski’s inductive definitions of the satisfaction, and truth, of the formulas of PA under the above interpretation. Moreover, its inclusion makes $\mathcal{I}_{PA(N,\ Standard)}$ extraneously non-finitary [36].

We note further that if PA is $\omega$inconsistent, then Aristotle’s particularisation does not hold over $N$, and the interpretation $\mathcal{I}_{PA(N,\ Standard)}$ is not sound over $N$.

4.2 An instantiational interpretation of PA over the domain $\mathbb {N}$ of the PA numerals

We next consider an instantiational interpretation $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ of PA over the domain $\mathbb {N}$ of the PA numerals [37] where:

(a) we define S as PA with standard first-order predicate calculus as the underlying logic;

(b) we define $D$ as the set $\mathbb {N}$ of PA numerals;

(c) for any atomic formula $[A(x_{1}, x_{2}, \ldots, x_{n})]$ of PA and any sequence $[(a_{1}, a_{2}, \ldots, a_{n})]$ of PA numerals in $\mathbb {N}$, we take $\|$SATCON($\mathcal{I}_{PA(\mathbb{N})}$)$\|$ as:

$\|$$[A(a_{1}, a_{2}, \ldots, a_{n})]$ is provable in PA and, for any given sequence of numerals $[(b_{1}, b_{2}, \ldots, b_{n})]$ of PA, the formula $[A(b_{1}, b_{2}, \ldots, b_{n})]$ is decidable as either provable or not provable in PA$\|$;

(d) we define the witness $\mathcal{W}_{(\mathbb {N},\ Instantiational)}$ as the meta-theory $\mathcal{M}_{PA}$ of PA.

Lemma 2: $[A(x_{1}, x_{2}, \ldots, x_{n})]$ is always algorithmically verifiable in PA by $\mathcal{W}_{(\mathbb {N},\ Instantiational)}$.

Proof: It follows from Gödel’s definition of the primitive recursive relation $xBy$ [38]—where $x$ is the Gödel number of a proof sequence in PA whose last term is the PA formula with Gödel-number $y$—that, if $[A(x_{1}, x_{2}, \ldots, x_{n})]$ is an atomic formula of PA, $\mathcal{M}_{PA}$ can algorithmically verify for any given sequence $[(b_{1}, b_{2}, \ldots, b_{n})]$ of PA numerals which one of the PA formulas $[A(b_{1}, b_{2}, \ldots, b_{n})]$ and $[\neg A(b_{1}, b_{2}, \ldots, b_{n})]$ is necessarily PA-provable. $\Box$

Now, if PA is consistent but not $\omega$-consistent, then there is a Gödelian formula $[R(x)]$ [39] such that:

(i) $[(\forall x)R(x)]$ is not PA-provable;

(ii) $[\neg (\forall x)R(x)]$ is PA-provable;

(iii) for any given numeral $[n]$, the formula $[R(n)]$ is PA-provable.

However, if $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ is sound over $\mathbb {N}$, then (ii) implies contradictorily that it is not the case that, for any given numeral $[n]$, the formula $[R(n)]$ is PA-provable.

It follows that if $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ is sound over $\mathbb {N}$, then PA is $\omega$-consistent and, ipso facto, Aristotle’s particularisation must hold over $N$.

Moreover, if PA is consistent, then every PA-provable formula interprets as true under some sound interpretation of PA over $N$. Hence $\mathcal{M}_{PA}$ can effectively decide whether, for any given sequence of natural numbers $(b_{1}^{*}, b_{2}^{*},$ $\ldots, b_{n}^{*})$ in $N$, the proposition $A^{*}(b_{1}^{*}, b_{2}^{*}, \ldots, b_{n}^{*})$ holds or not in $N$.

It follows that $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ can be viewed as a constructive formalisation of the standard’ interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of PA in which we do not need to non-constructively assume that Aristotle’s particularisation holds over $N$.

4.3 An algorithmic interpretation of PA over the domain $N$ of the natural numbers

We finally consider the purely algorithmic interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA over the domain $N$ of the natural numbers where:

(a) we define S as PA with standard first-order predicate calculus as the underlying logic;

(b) we define $D$ as the set $N$ of natural numbers;

(c) for any atomic formula $[A(x_{1}, x_{2}, \ldots, x_{n})]$ of PA and any sequence $(a_{1}, a_{2}, \ldots, a_{n})$ of natural numbers in $N$, we take $\|$SATCON($\mathcal{I}_{PA(N)}$)$\|$ as:

$\|$$A^{*}(a_{1}^{*}, a_{2}^{*}, \ldots, a_{n}^{*})$ holds in $N$ and, for any given sequence $(b_{1}^{*}, b_{2}^{*}, \ldots, b_{n}^{*})$ of $N$, the proposition $A^{*}(b_{1}^{*}, b_{2}^{*}, \ldots, b_{n}^{*})$ is decidable as either holding or not holding in $N$$\|$;

(d) we define the witness $\mathcal{W}_{(N,\ Algorithmic)}$ as any simple functional language that gives evidence that $\|$SATCON($\mathcal{I}_{PA(N)}$)$\|$ is always effectively decidable in $N$:

Lemma 3: $A^{*}(x_{1}, x_{2}, \ldots, x_{n})$ is always algorithmically computable in $N$ by $\mathcal{W}_{(N,\ Algorithmic)}$.

Proof: If $[A(x_{1}, x_{2}, \ldots, x_{n})]$ is an atomic formula of PA then, for any given sequence of numerals $[b_{1}, b_{2}, \ldots, b_{n}]$, the PA formula $[A(b_{1}, b_{2},$ $\ldots, b_{n})]$ is an atomic formula of the form $[c=d]$, where $[c]$ and $[d]$ are atomic PA formulas that denote PA numerals. Since $[c]$ and $[d]$ are recursively defined formulas in the language of PA, it follows from a standard result [40] that, if PA is consistent, then $[c=d]$ is algorithmically computable as either true or false in $N$. In other words, if PA is consistent, then $[A(x_{1}, x_{2}, \ldots, x_{n})]$ is algorithmically computable (since there is an algorithm that, for any given sequence of numerals $[b_{1}, b_{2}, \ldots, b_{n}]$, will give evidence whether $[A(b_{1}, b_{2},$ $\ldots, b_{n})]$ interprets as true or false in $N$. The lemma follows. $\Box$

It follows that $\mathcal{I}_{PA(N,\ Algorithmic)}$ is an algorithmic formulation of the standard’ interpretation of PA over $N$ in which we do not extraneously assume either that Aristotle’s particularisation holds over $N$ or, equivalently, that PA is $\omega$-consistent.

5 Formally defining the standard interpretation of PA over $N$ constructively

It follows from the analysis of the applicability of Tarski’s inductive definitions of satisfiability’ and truth’ in Section 4 that we can formally define the standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of PA constructively where:

(a) we define S as PA with standard first-order predicate calculus as the underlying logic;

(b) we define $D$ as $N$;

(c) we take SM($\mathcal{I}_{PA(N,\ Standard)}$) as any simple functional language.

We note that:

Theorem 2: The atomic formulas of PA are algorithmically verifiable under the standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$.

Proof: If $[A(x_{1}, x_{2}, \ldots, x_{n})]$ is an atomic formula of PA then, for any given denumerable sequence of numerals $[b_{1}, b_{2}, \ldots]$, the PA formula $[A(b_{1}, b_{2},$ $\ldots, b_{n})]$ is an atomic formula of the form $[c=d]$, where $[c]$ and $[d]$ are atomic PA formulas that denote PA numerals. Since $[c]$ and $[d]$ are recursively defined formulas in the language of PA, it follows from a standard result that, if PA is consistent, then $[c=d]$ interprets as the proposition $c=d$ which either holds or not for a witness $\mathcal{W}_{N}$ in $N$.

Hence, if PA is consistent, then $[A(x_{1}, x_{2}, \ldots, x_{n})]$ is algorithmically verifiable since, for any given denumerable sequence of numerals $[b_{1}, b_{2}, \ldots]$, we can define an algorithm that provides evidence that the PA formula $[A(b_{1}, b_{2}, \ldots, b_{n})]$ is decidable under the interpretation.

The theorem follows. $\Box$

It immediately follows that:

Corollary 1: The satisfaction’ and truth’ of PA formulas containing logical constants can be defined under the standard interpretation of PA over $N$ in terms of the evidence provided by the computations of a simple functional language.

Corollary 2: The PA-formulas are decidable under the standard interpretation of PA over $N$ if, and only if, they are algorithmically verifiable under the interpretation.

5.1 Defining algorithmic truth’ under the standard interpretation of PA over $N$

Now we note that, in addition to Theorem 2:

Theorem 3: The atomic formulas of PA are algorithmically computable under the standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$.

Proof: If $[A(x_{1}, x_{2}, \ldots, x_{n})]$ is an atomic formula of PA then we can define an algorithm that, for any given denumerable sequence of numerals $[b_{1}, b_{2}, \ldots]$, provides evidence whether the PA formula $[A(b_{1}, b_{2}, \ldots, b_{n})]$ is true or false under the interpretation.

The theorem follows. $\Box$

This suggests the following definitions:

Definition 9: A well-formed formula $[A]$ of PA is algorithmically true under $\mathcal{I}_{PA(N,\ Standard)}$ if, and only if, there is an algorithm which provides evidence that, given any denumerable sequence $t$ of $N$, $t$ satisfies $[A]$;

Definition 10:A well-formed formula $[A]$ of PA is algorithmically false under $\mathcal{I}_{PA(N,\ Standard)}$ if, and only if, it is not algorithmically true under $\mathcal{I}_{PA(N)}$.

5.2 The PA axioms are algorithmically computable

The significance of defining algorithmic truth’ under $\mathcal{I}_{PA(N,\ Standard)}$ as above is that:

Lemma 4: The PA axioms PA$_{1}$ to PA$_{8}$ are algorithmically computable as algorithmically true over $N$ under the interpretation $\mathcal{I}_{PA(N,\ Standard)}$.

Proof: Since $[x+y]$, $[x \star y]$, $[x = y]$, $[{x^{\prime}}]$ are defined recursively [41], the PA axioms PA$_{1}$ to PA$_{8}$ interpret as recursive relations that do not involve any quantification. The lemma follows straightforwardly from Definitions 3 to 8 in Section 4 and Theorem 2. $\Box$

Lemma 5: For any given PA formula $[F(x)]$, the Induction axiom schema $[F(0)$ $\rightarrow (((\forall x)(F(x) \rightarrow F(x^{\prime}))) \rightarrow (\forall x)F(x))]$ interprets as algorithmically true under $\mathcal{I}_{PA(N,\ Standard)}$.

Proof: By Definitions 3 to 10:

(a) If $[F(0)]$ interprets as algorithmically false under $\mathcal{I}_{PA(N,\ Standard)}$ the lemma is proved.

Since $[F(0) \rightarrow (((\forall x)(F(x) \rightarrow F(x^{\prime}))) \rightarrow (\forall x)F(x))]$ interprets as algorithmically true if, and only if, either $[F(0)]$ interprets as algorithmically false or $[((\forall x)(F(x) \rightarrow F(x^{\prime}))) \rightarrow (\forall x)F(x)]$ interprets as algorithmically true.

(b) If $[F(0)]$ interprets as algorithmically true and $[(\forall x)(F(x) \rightarrow F(x^{\prime}))]$ interprets as algorithmically false under $\mathcal{I}_{PA(N,\ Standard)}$, the lemma is proved.

(c) If $[F(0)]$ and $[(\forall x)(F(x) \rightarrow F(x^{\prime}))]$ both interpret as algorithmically true under $\mathcal{I}_{PA(N,\ Standard)}$, then by Definition 9 there is an algorithm which, for any natural number $n$, will give evidence that the formula $[F(n) \rightarrow F(n^{\prime})]$ is true under $\mathcal{I}_{PA(N,\ Standard)}$.

Since $[F(0)]$ interprets as algorithmically true under $\mathcal{I}_{PA(N,\ Standard)}$, it follows that there is an algorithm which, for any natural number $n$, will give evidence that the formula $[F(n)]$ is true under the interpretation.

Hence $[(\forall x)F(x)]$ is algorithmically true under $\mathcal{I}_{PA(N,\ Standard)}$.

Since the above cases are exhaustive, the lemma follows. $\Box$

The Poincaré-Hilbert debate: We note that Lemma 5 appears to settle the Poincaré-Hilbert debate [42] in the latter’s favour. Poincaré believed that the Induction Axiom could not be justified finitarily, as any such argument would necessarily need to appeal to infinite induction. Hilbert believed that a finitary proof of the consistency of PA was possible.

Lemma 6: Generalisation preserves algorithmic truth under $\mathcal{I}_{PA(N,\ Standard)}$.

Proof: The two meta-assertions:

$[F(x)]$ interprets as algorithmically true under $\mathcal{I}_{PA(N,\ Standard)}$ [43]

and

$[(\forall x)F(x)]$ interprets as algorithmically true under $\mathcal{I}_{PA(N,\ Standard)}$

both mean:

$[F(x)]$ is algorithmically computable as always true under $\mathcal{I}_{PA(N),}$ $_{Standard)}$. $\Box$

It is also straightforward to see that:

Modus Ponens preserves algorithmic truth under $\mathcal{I}_{PA(N,\ Standard)}$. $\Box$

We thus have that:

Theorem 4: The axioms of PA are always algorithmically true under the interpretation $\mathcal{I}_{PA(N,\ Standard)}$, and the rules of inference of PA preserve the properties of algorithmic satisfaction/truth under $\mathcal{I}_{PA(N,\ Standard)}$ [44]. $\Box$

5.3 The interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA over $N$ is sound

We conclude from Section 4.3 and Section 5.2 that there is an algorithmic interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA over $N$ such that:

Theorem 5: The interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA is sound over $N$.

Proof: It follows immediately from Theorem 4 that the axioms of PA are always true under the interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$, and the rules of inference of PA preserve the properties of satisfaction/truth under $\mathcal{I}_{PA(N,\ Algorithmic)}$. $\Box$

We thus have a finitary proof that:

Theorem 6: PA is consistent. $\Box$

Conclusion

We have shown that although conventional wisdom is justified in assuming that the quantified arithmetical propositions of the first order Peano Arithmetic PA are constructively decidable under the standard interpretation of PA over the domain $N$ of the natural numbers, the assumption does not address—and implicitly conceals—a significant ambiguity that needs to be made explicit.

Reason: Tarski’s inductive definitions admit evidence-based interpretations of the first-order Peano Arithmetic PA that allow us to define the satisfaction and truth of the quantified formulas of PA constructively over $N$ in two essentially different ways.

First in terms of algorithmic verifiabilty. We show that this allows us to define a formal instantiational interpretation $\mathcal{I}_{PA(\mathbb {N},\ Instantiational)}$ of PA over the domain $\mathbb {N}$ of the PA numerals that is sound (i.e. PA theorems interpret as true in $N$) if, and only if, the standard interpretation of PA over $N$—which is not known to be finitary—is sound.

Second in terms of algorithmic computability. We show that this allows us to define a finitary algorithmic interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA over $N$ which is sound, and so we may conclude that PA is consistent.

Acknowledgements

We would like to thank Professor Rohit Parikh for his suggestion that this paper should appeal to the computations of a simple functional language in general, and avoid appealing to the computations of a Turing machine in particular.

References

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Br13 L. E. J. Brouwer. 1913. Intuitionism and Formalism. Inaugural address at the University of Amsterdam, October 14, 1912. Translated by Professor Arnold Dresden for the Bulletin of the American Mathematical Society, Volume 20 (1913), pp.81-96. 1999. Electronically published in Bulletin (New Series) of the American Mathematical Society, Volume 37, Number 1, pp.55-64.

Co66 Paul J. Cohen. 1966. Set Theory and the Continuum Hypothesis. (Lecture notes given at Harvard University, Spring 1965) W. A. Benjamin, Inc., New York.

Cr05 John N. Crossley. 2005. What is Mathematical Logic? A Survey. Address at the First Indian Conference on Logic and its Relationship with Other Disciplines held at the Indian Institute of Technology, Powai, Mumbai from January 8 to 12. Reprinted in Logic at the Crossroads: An Interdisciplinary View – Volume I (pp.3-18). ed. Amitabha Gupta, Rohit Parikh and Johan van Bentham. 2007. Allied Publishers Private Limited, Mumbai.

Da82 Martin Davis. 1958. Computability and Unsolvability. 1982 ed. Dover Publications, Inc., New York.

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Fe06 Solomon Feferman. 2006. Are There Absolutely Unsolvable Problems? Gödel’s Dichotomy. Philosophia Mathematica (2006) 14 (2): 134-152.

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Go31 Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York.

HA28 David Hilbert & Wilhelm Ackermann. 1928. Principles of Mathematical Logic. Translation of the second edition of the Grundzüge Der Theoretischen Logik. 1928. Springer, Berlin. 1950. Chelsea Publishing Company, New York.

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Hi27 David Hilbert. 1927. The Foundations of Mathematics. Text of an address delivered in July 1927 at the Hamburg Mathematical Seminar. In Jean van Heijenoort. 1967.Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts.

Kl52 Stephen Cole Kleene. 1952. Introduction to Metamathematics. North Holland Publishing Company, Amsterdam.

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Notes

Return to 1: For a brief recent review of such challenges, see Fe06, Fe08.

Return to 2: As detailed in Section 4.

Return to 3: We take this to be the first-order theory S defined in Me64, p.102.

Return to 4: We essentially follow the definitions in Me64, p.49.

Return to 5: We define this important concept explicitly later in Section 2.1. Loosely speaking, Aristotle’s particularisation is the assumption that we may always interpret the formal expression [$(\exists x)F(x)$]’ of a formal language under an interpretation as There exists an object $s$ in the domain of the interpretation such that $F(s)$‘.

Return to 6: See Me64, p.107.

Return to 7: Finitary’ in the sense that “… there should be an algorithm for deciding the truth or falsity of any mathematical statement“. For a brief review of finitism’ and constructivity’ in the context of this paper see Fe08.

Return to 8: Section 3, Definition 1.

Return to 9: Section 3, Definition 2.

Return to 10: The possibility/impossibility of such justification was the subject of the famous Poincaré-Hilbert debate. See Hi27, p.472; also Br13, p.59; We27, p.482; Pa71, p.502-503.

Return to 11: In the sense highlighted by Elliott Mendelson in Me64, p.261.

Return to 12: cf. Me64, p258.

Return to 13: See for instance http://en.wikipedia.org/wiki/Hilbert’s\_program.

Return to 14: Section 5.2, Theorem 4.

Return to 15: Section 5.3, Theorem 5.

Return to 16: Section 5.3, Theorem 6.

Return to 17: Formal definitions are given in Section 4.

Return to 18: Mu91.

Return to 19: Such as, for instance, that of a deterministic Turing machine (Me64, pp.229-231) based essentially on Alan Turing’s seminal 1936 paper on computable numbers (Tu36).

Return to 20: Ta33.

Return to 21: As, for instance, in Go31.

Return to 22: Essentially reflecting Brouwer’s objection to the assumption of Aristotle’s particularisation over an infinite domain.

Return to 23: An assumption explicitly introduced by Gödel in Go31.

Return to 24: HA28, pp.58-59.

Return to 25: See Hi25, p.382; HA28, p.48; Sk28, p.515; Go31, p.32.; Kl52, p.169; Ro53, p.90; BF58, p.46; Be59, pp.178 & 218; Su60, p.3; Wa63, p.314-315; Qu63, pp.12-13; Kn63, p.60; Co66, p.4; Me64, p.52(ii); Nv64, p.92; Li64, p.33; Sh67, p.13; Da82, p.xxv; Rg87, p.xvii; EC89, p.174; Mu91; Sm92, p.18, Ex.3; BBJ03, p.102; Cr05, p.6.

Return to 26: Br08.

Return to 27: Go31, p.23 and p.28.

Return to 28: In his introduction on p.9 of Go31.

Return to 29: The distinction sought to be made between algorithmic verifiabilty and algorithmic computability can be viewed as reflecting in number theory the similar distinction in analysis between, for instance, continuous functions (Ru53, p.65, $\S$4.5) and uniformly continuous functions (Ru53, p.65, $\S$4.13); or that between convergent sequences (Ru53, p.65, $\S$7.1) and uniformly convergent sequences (Ru53, p.65, $\S$7.7).}

Return to 30: cf. Me64, p.51.

Return to 31: In the sense of Mu91.

Return to 32: See Me64, p.51; Mu91.

Return to 33: cf. Me64, pp.51-53.

Return to 34: Where the string $[(\exists \ldots)]$ is defined as—and is to be treated as an abbreviation for—the string $[\neg (\forall \ldots) \neg]$. We do not consider the case where the underlying logic is Hilbert’s formalisation of Aristotle’s logic of predicates in terms of his $\epsilon$-operator (Hi27, pp.465-466).

Return to 35: Hence a PA formula such as $[(\exists x)F(x)]$ interprets under $\mathcal{I}_{PA(N,\ Standard)}$ as There is some natural number $n$ such that $F(n)$ holds in $N$.

Return to 36: Br08.

Return to 37: The raison d’être, and significance, of such interpretation is outlined in this short unpublished note accessible at http://alixcomsi.com/8\_Meeting\_Wittgenstein\_requirement\_1000.pdf.

Return to 38: Go31, p. 22(45).

Return to 39: Gödel constructively defines, and refers to, this formula by its Gödel number $r$‘: see Go31, p.25, Eqn.(12).

Return to 40: For any natural numbers $m,\ n$, if $m \neq n$, then PA proves $[\neg(m = n)]$ (Me64, p.110, Proposition 3.6). The converse is obviously true.

Return to 41: cf. Go31, p.17.

Return to 42: See Hi27, p.472; also Br13, p.59; We27, p.482; Pa71, p.502-503.

Return to 43: See Definition 7.

Return to 44: Without appeal, moreover, to Aristotle’s particularisation.

See also (i) this later publication by Sebastian Grève, where he concludes that “… while Gödel indeed showed some significant understanding of Wittgenstein here, ultimately, Wittgenstein perhaps understood Gödel better than Gödel understood himself”; and (ii) this note on Rosser’s Rule C and Wittgenstein’s objections on purely philosophical considerations to Gödel’s reasoning and conclusions, where we show that, although not at all obvious (perhaps due to Gödel’s overpoweringly plausible presentation of his interpretation of his own formal reasoning over the years) what Gödel claimed to have proven is not—as suspected and held by Wittgenstein—supported by Gödel’s formal argumentation.

A: Misunderstanding Gödel’s Theorems VI and XI: Incompleteness and Undecidability

In an informal essay, “Misunderstanding Gödel’s Theorems VI and XI: Incompleteness and Undecidability“, DPhil candidate Sebastian Grève at The Queen’s College, Oxford, attempts to come to terms with what he subjectively considers:

“… has not been properly addressed as such by philosophers hitherto as of great philosophical importance in our understanding of Gödel’s Incompleteness Theorems.”

Grève’s is an unusual iconoclastic perspective:

“This essay is an open enquiry towards a better understanding of the philosophical significance of Gödel’s two most famous theorems. I proceed by a discussion of several common misunderstandings, led by the following four questions:

1) Is the Gödel sentence true?

2) Is the Gödel sentence undecidable?

3) Is the Gödel sentence a statement?

4) Is the Gödel sentence a sentence?

Asking these questions in this order means to trace back the steps of Gödel’s basic philosophical interpretation of his formal results. What I call the basic philosophical interpretation is usually just taken for granted by philosopher’s writing about Gödel’s theorems.”

In a footnote Grève acknowledges Wittgenstein’s influence by suggesting that:

“This essay can be read as something like a free-floating interpretation of the theme of Wittgenstein’s remarks on Gödel’s Incompleteness Theorems in Wittgenstein: 1978[RFM], I-(III), partly following Floyd: 1995 but especially Kienzler: 2008, and constituting a reply to inter alia Rodych: 2003”.

B: Why we may see the trees, but not the forest

We note that Grève’s four points are both overdue and well-made:

1. Is the Gödel sentence true?

Grève’s objection that standard interpretations are obscure when they hold the Gödel sentence as being intuitively true deserves consideration (see this post).

The ‘truth’ of the sentence should and does—as Wittgenstein stressed and suggested—follow objectively from the axioms and rules of inference of arithmetic.

2. Is the Gödel sentence undecidable?

Grève’s observation that the ‘undecidability’ of the Gödel sentence conceals a philosophically questionable assumption is well-founded.

The undecidability in question follows only on the assumption of ‘$\omega$-consistency’ made explicitly by Gödel.

This assumption is actually logically equivalent to the philosophically questionable assertion that from the provability of $[\neg(\forall x)R(x)]$ we may conclude the existence of some numeral $[n]$ for which $[R(n)]$ is provable.

Since Rosser’s proof implicitly makes this assumption by means of his logically questionable Rule C, his claim of avoiding omega-consistency for arithmetic is illusory.

3. Is the Gödel sentence a statement?

Grève rightly holds that the Gödel sentence should be treated as a valid statement within the formal arithmetic S, since it is structurally defined as a well-formed formula of S.

4. Is the Gödel sentence a sentence?

Grève’s concern about whether the Gödel sentence of S is a valid arithmetical proposition under interpretation also seems to need serious philosophical consideration.

It can be argued (see the comment following the proof of Lemma 9 of this preprint) that the way the sentence is formally defined as the universal quantification of an instantiationally (but not algorithmically) defined arithmetical predicate does not yield an unequivocally defined arithmetical proposition in the usual sense under interpretation.

In this post [*] we shall not only echo Grève’s disquietitude, but argue further that Gödel’s interpretation and assessment of his own formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions is, essentially, a post-facto imposition that continues to influence standard expositions of Gödel’s reasoning misleadingly.

Feynman’s cover-up factor

Our thesis is influenced by physicist Richard P. Feynman, who started his 1965 Nobel Lecture with a penetrating observation:

We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn’t any place to publish, in a dignified manner, what you actually did in order to get to do the work.

That such cover up’ may have unintended—and severely limiting—consequences on a discipline is suggested by Gödel’s interpretation, and assessment, of his own formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions (Go31).

Thus, in his informal preamble to the result that he intended to prove formally, Gödel wrote (cf. Go31, p.9):

The analogy of this result with Richard’s antinomy is immediately evident; there is also a close relationship with the Liar Paradox … Thus we have a proposition before us which asserts its own unprovability.

Further, interpreting the significance of his formal reasoning as having established the existence of a formally undecidable arithmetical proposition that is, however, decidable by meta-mathematical arguments, Gödel noted that:

The precise analysis of this remarkable circumstance leads to surprising results concerning consistency proofs of formal systems … (Go31, p.9)

The true reason for the incompleteness which attaches to all formal systems of mathematics lies, as will be shown in Part II of this paper, in the fact that the formation of higher and higher types can be continued into the transfinite (c.f., D. Hilbert, Über das Unendliche’, Math. Ann. 95, p. 184), while, in every formal system, only countable many are available. Namely, one can show that the undecidable sentences which have been constructed here always become decidable through adjunction of suitable high types (e.g. of the type $\omega$ to the system $P$. A similar result also holds for the axiom systems of set theory. (Go31, p.28, footnote 48a)

The explicit thesis of this foundational paper is that the above interpretation is an instance of a cover up’—in Feynman’s sense—which appears to be a post-facto imposition that, first, continues to echo in and misleadingly [1] influence standard expositions of Gödel’s reasoning when applied to a first-order Peano Arithmetic, PA, and, second, that it obscures the larger significance of the genesis of Gödel’s reasoning.

As Gödel’s various remarks in Go31 suggest, this possibly lay in efforts made at the dawn of the twentieth century—largely as a result of Brouwer’s objections (Br08)—to define unambiguously the role that the universal and existential quantifiers played in formal mathematical reasoning.

That this issue is critical to Gödel’s reasoning in Go31, but remains unresolved in it, is obscured by his powerful presentation and interpretation.

So, to grasp the underlying mathematical significance of Gödel’s reasoning, and of what he has actually achieved, one may need to avoid focusing (as detailed in the previous posts on A foundational perspective on the semantic and logical paradoxes; in this post on undecidable Gödelian propositions, and in this preprint on undecidable Gödelian propositions):

$\bullet$ on the analogy of the so-called Liar paradox’;

$\bullet$ on Gödel’s interpretation of his arithmetical proposition as asserting its own formal unprovability in his formal Peano Arithmetic P (Go31, pp.9-13);

$\bullet$ on his interpretation of the reasons for the incompleteness’ of P; and

$\bullet$ on his assessment and interpretation of the formal consequences of such incompleteness’.

We show in this paper that, when applied to PA [2], all of these obscure the deeper significance of what Gödel actually achieved in Go31.

C: Hilbert: If the $\omega$-Rule is true, can P be completed?

Instead, Gödel’s reasoning may need to be located specifically in the context of Hilbert’s Program (cf. Hi30, pp.485-494) in which he proposed an $\omega$-rule as a finitary means of extending a Peano Arithmetic—such as his formal system P in Go31—to a possible completion (i.e. to logically showing that, given any arithmetical proposition, either the proposition, or its negation, is formally provable from the axioms and rules of inference of the extended Arithmetic).

Hilbert’s $\omega$-Rule: If it is proved that the P-formula [$F(x)$] interprets as a true numerical formula for each given P-numeral [$x$], then the P-formula $[(\forall x)F(x)]$ may be admitted as an initial formula (axiom) in P.

It is likely that Gödel’s 1931 paper evolved out of attempts to prove Hilbert’s $\omega$-rule in the limited—and more precise—sense that if a formula [$F(n)$] is provable in P for each given numeral [$n$], then the formula [$(\forall x)F(x)$] must be provable in P.

Now, if we meta-assume Hilbert’s $\omega$-rule for P, then it follows that, if P is consistent, then there is no P-formula [$F(x)$] for which, first, [$\neg(\forall x)F(x)$] is P-provable and, second, [$F(n)$] is P-provable for any given P-numeral [$n$].

Gödel defined a consistent Peano Arithmetic with the above property as additionally $\omega$-consistent (Go31, pp.23-24).

D: The significance of $\omega$-consistency

To place the significance of $\omega$-consistency in a current perspective, we note that the standard model of the first order Peano Arithmetic PA (cf. Me64, p.107; Sc67, p.23, p.209; BBJ03, p.104) presumes [3] that the standard interpretation M of PA (under which the PA-formula [$(\exists x)R(x)$], which is merely an abbreviation for $[\neg(\forall x)\neg R(x)]$, interprets as true if, and only if, $R(n)$ holds for some natural number $n$ under M) is sound (cf. BBJ03, p.174).

Clearly, if such an interpretation of the existential quantifier is sound, it immediately implies that PA is necessarily $\omega$-consistent [4].

Since Brouwer’s main objection was to Hilbert’s presumption that such an interpretation of the existential quantifier is sound, Gödel explicitly avoided this assumption in his seminal 1931 paper (Go31, p.9) in order to ensure that his reasoning was acceptable as “constructive” and “intuitionistically unobjectionable” (Go31, p.26).

He chose, instead, to present the formal undecidability of his arithmetical proposition—and the consequences arising from it—as explicitly conditional on the assumption of the formal property of $\omega$-consistency for his Peano Arithmetic P under the unqualified—and, as we show below, mistaken—belief that:

PA is $\omega$-consistent (Go31, p.28, footnote 48a).

E: Gödel: If the $\omega$-Rule is true, P cannot be completed

Now, Gödel’s significant achievement in Go31 was the discovery that, if P is consistent, then it was possible to construct a P-formula, [$R(x)$] [5], such that $[R(n)]$ is P-provable for any given P-numeral [$n$] (Go31, p.25(2)), but [$(\forall x)R(x)$] is P-unprovable (Go31, p.25(1)).

However, it becomes apparent from his remarks in Go31 that Gödel considered his more significant achievement the further argument that, if P is assumed $\omega$-consistent, then both [$(\forall x)R(x)$] and [$\neg (\forall x)R(x)$] [6] are P-unprovable, and so P is incomplete!

This is the substance of Gödel’s Theorem VI (Go31, p.24).

Although this Theorem neither validated nor invalidated Hilbert’s $\omega$-rule, it did imply that assuming the rule led not to the completion of a Peano Arithmetic as desired by Hilbert, but to its essential incompletability!

F: The $\omega$-Rule is inconsistent with PA

Now, apparently, the possibility neither considered by Gödel in 1931, nor seriously since, is that a formal sytem of Peano Arithmetic—such as PA—may be consistent and $\omega$inconsistent.

If so, one would ascribe this omission to the cover up’ factor mentioned by Feynman, since a significant consequence of Gödel’s reasoning—in the first half of his proof of his Theorem VI—is that it actually establishes PA as $\omega$inconsistent (as detailed in Corollary 9 of this preprint and Corollary 4 of this post).

In other words, we can logically show for Gödel’s formula [$R(x)$] that [$\neg(\forall x)$ $R(x)$] is PA-provable, and that [$R(n)$] is PA-provable for any given PA-numeral [$n$].

Consequently, Gödel’s Theorem VI is vacuously true for PA, and it also follows that Hilbert’s $\omega$-Rule is inconsistent with PA!

G: Need: A paradigm shift in interpreting the quantifiers

Thus Gödel’s unqualified belief that:

PA is $\omega$-consistent

was misplaced, and Brouwer’s objection to Hilbert’s presumption—that the above interpretation of the existential quantifier is sound—was justified; since, if PA is consistent, then it is provably $\omega$inconsistent, from which it follows that the standard interpretation M of PA is not sound.

Hence we can no longer interpret [$\neg(\forall x)F(x)$] is true’ maximally under the standard interpretation of PA as:

(i) The arithmetical relation $F(n)$ is not always [7] true.

However, since the theorems of PA—when treated as Boolean functions—are Turing-computable as always true under a sound finitary interpretation $\beta$ of PA, we can interpret [$\neg(\forall x)F(x)$] is true’ minimally as:

(ii) The arithmetical relation $F(n)$ is not Turing-computable as always true.

This interpretation allows us to conclude from Gödel’s meta-mathematical argument that we can construct a PA-formula [$(\forall x)R(x)$] that is unprovable in PA, but which is true under a sound interpretation of PA [8] although we may now no longer conclude from Gödel’s reasoning that there is an undecidable arithmetical PA-proposition.

Moreover, the interpretation admits an affirmative answer to Hilbert’s query: Is PA complete or completeable?

H: PA is algorithmically complete

In outline, the basis from which this conclusion follows formally is that:

(i) Gödel’s proof of the essential incompleteness of any formal system of Peano Arithmetic (Go31, Theorem VI, p.24) explicitly assumes that the arithmetic is $\omega$-consistent;

(ii) Rosser’s extension of Gödel’s proof of the essential incompleteness of any formal system of Peano Arithmetic (cf. Ro36, Theorem II, p.233) implicitly presumes that the Arithmetic is $\omega$-consistent (as detailed in this post);

(iii) PA is $\omega$inconsistent (as detailed in Corollary 9 of this preprint);

(iv) The classical standard’ interpretation of PA (cf. Me64, section \S 2, pp.49-53; p107) over the structure [$N$]—defined as {$N$ (the set of natural numbers); $=$ (equality); $'$ (the successor function); $+$ (the addition function); $\ast$ (the product function); $0$ (the null element)}— does not define a finitary model of PA (as detailed in the paper titled Evidence-Based Interpretations of PA presented at IACAP/AISB Turing 2012, Birmingham, UK in July 2012);

(v) We can define a sound interpretation $\beta$ of PA—in terms of Turing-computability—which yields a finitary model of PA, but which does not admit a non-standard model for PA (as detailed in this paper);

(vi) PA is algorithmically complete in the sense that an arithmetical proposition $F$ defines a Turing-machine TM$_{F}$ which computes $F$ as true under $\beta$ if, and only if, the corresponding PA-formula [$F$] is PA-provable (as detailed in Section 8 of this preprint).

I: Gödel’s proof of his Theorem XI does not withstand scrutiny

Since Gödel’s proof of his Theorem XI (Go31, p.36)—in which he claims to show that the consistency of his formal system of Peano Arithmetic P can be expressed as a P-formula which is not provable in P—appeals critically to his Theorem VI, it follows that this proof cannot be applied to PA.

However, we show below that there are other, significant, reasons why Gödel’s reasoning in this proof must be treated as classically objectionable per se.

J: Why Gödel’s interpretation of the significance of his Theorem XI is classically objectionable

Now, in his Theorem XI, Gödel constructs a formula [$W$] [9] in P and assumes that [$W$] translates—under a sound interpretation of P—as an arithmetical proposition that is true if, and only if, a specified formula of P is unprovable in P.

Now, if there were such a P-formula, then, since an inconsistent system necessarily proves every well-formed formula of the system, it would follow that a proof sequence within P proves that P is consistent.

However, Gödel shows that his formula [$W$] is not P-provable (Go31, p.37).

He concludes that the consistency of any formal system of Peano Arithmetic is not provable within the Arithmetic. [10]

K: Defining meta-propositions of P arithmetically

Specifically, Gödel first shows how 46 meta-propositions of P can be defined by means of primitive recursive functions and relations (Go31, pp.17-22).

These include:

($\#23$) A primitive recursive relation, Form($x$), which is true if, and only if, $x$ is the Gödel-number of a formula of P;

($\#45$) A primitive recursive relation, $xBy$, which is true if, and only if, $x$ is the Gödel-number of a proof sequence of P whose last formula has the Gödel-number $y$.

Gödel assures the constructive nature of the first 45 definitions by specifying (cf. Go31, p.17, footnote 34):

Everywhere in the following definitions where one of the expressions $\forall x$‘, $\exists x$‘, $\epsilon x$ (There is a unique $x$)’ occurs it is followed by a bound for $x$. This bound serves only to assure the recursive nature of the defined concept.

Gödel then defines a meta-mathematical proposition that is not recursive:

($\#46$) A proposition, $Bew(x)$, which is true if, and only if, $(\exists y)yBx$ is true.

Thus $Bew(x)$ is true if, and only if, $x$ is the Gödel-number of a provable formula of P.

L: Expressing arithmetical functions and relations in P

Now, by Gödel’s Theorem VII (Go31, p.29), any recursive relation, say $Q(x)$, can be represented in P by some, corresponding, arithmetical formula, say [$R(x)$], such that, for any natural number $n$:

If $Q(n)$ is true, then [$R(n)$] is P-provable;

If $Q(n)$ is false, then [$\neg R(n)$] is P-provable.

However, Gödel’s reasoning in the first half of his Theorem VI (Go31, p.25(1)) establishes that the above representation does not extend to the closure of a recursive relation, in the sense that we cannot assume:

If $(\forall x)Q(x)$ is true (i.e, $Q(n)$ is true for any given natural number), then $[(\forall x)R(x)]$ is P-provable.

In other words, we cannot assume that, even though the recursive relation $Q(x)$ is instantiationally equivalent to a sound interpretation of the P-formula [$R(x)$], the number-theoretic proposition $(\forall x)Q(x)$ must, necessarily, be logically equivalent to the—correspondingly sound—interpretation of the P-formula [$(\forall x)R(x)$].

The reason: In recursive arithmetic, the expression $(\exists x)F(x)$‘ is an abbreviation for the assertion:

(*) There is some (at least one) natural number $n$ such that $F(n)$ holds.

In a formal Peano Arithmetic, however, the formula [$(\exists x)F(x)$]’ is simply an abbreviation for [$\neg (\forall x)\neg F(x)$]’, which, under a sound finitary interpretation of the Arithmetic can have the verifiable translation:

(**) The relation $\neg F(x)$ is not Turing-computable as always true.

Moreover, Gödel’s Theorem VI establishes that we cannot conclude (*) from (**) without risking inconsistency.

Consequently, although a primitive recursive relation may be instantiationally equivalent to a sound interpretation of a P-formula, we cannot assume that the existential closure of the relation must have the same meaning as the interpretation of the existential closure of the corresponding P-formula.

However this, precisely, is the presumption made by Gödel in the proof of Theorem XI, from which he concludes that the consistency of P can be expressed in P, but is not P-provable.

M: Ambiguity in the interpreted meaning’ of formal mathematical expressions

The ambiguity in the meaning’ of formal mathematical expressions containing unrestricted universal and existential closure under an interpretation was emphasised by Wittgenstein (Wi56):

Do I understand the proposition “There is . . .” when I have no possibility of finding where it exists? And in so far as what I can do with the proposition is the criterion of understanding it … it is not clear in advance whether and to what extent I understand it.

N: Expressing “P is consistent” arithmetically

Specifically, Gödel defines the notion of “P is consistent” classically as follows:

P is consistent if, and only if, Wid(P) is true

where Wid(P) is defined as:

$( \exists x) (Form(x) \wedge \neg Bew(x))$

This translates as:

There is a natural number $n$ which is the Gödel-number of a formula of P, and this formula is not P-provable.

Thus, Wid(P) is true if, and only if, P is consistent.

O: Gödel: “P is consistent” is always expressible in P

However, Gödel, then, presumes that:

(i) Wid(P) can be represented by some formula [$W$] of P such that “[$W$] is true” and “Wid(P) is true” are logically equivalent (i.e., have the same meaning) under a sound interpretation of P;

(ii) if the recursive relation, $Q(x, p)$ (1931, p24(8.1)), is represented by the P-formula [$R(x, p)$], then the proposition “[$(\forall x)R(x, p)$] is true” is logically equivalent to (i.e., has the same meaning as) “$(\forall x)Q(x, p)$ is true” under a sound interpretation of P.

P: The loophole in Gödel’s presumption

Although, (ii), for instance, does follow if “[$(\forall x)R(x, p)$] is true” translates as “$R(x, p)$ is Turing-computable as always true”, it does not if “[$(\forall x)R(x, p)$] is true” translates as “$R(x, p)$ is constructively computable as true for any given natural number $n$, but it is not Turing-computable as true for any given natural number $n$“.

So, if [$W$], too, interprets as an arithmetical proposition that is constructively computable as true, but not Turing-computable as true, then the consistency of P may be provable instantiationally in P [11].

Hence, at best, Gödel’s reasoning can only be taken to establish that the consistency of P is not provable algorithmically in P.

Gödel’s broader conclusion only follows if P purports to prove its own consistency algorithmically.

However, Gödel’s particular argument, based on his definition of Wid(P), does not support this claim.

Bibliography

BBJ03 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic (4th ed). Cambridge University Press, Cambridge.

Br08 L. E. J. Brouwer. 1908. The Unreliability of the Logical Principles. English translation in A. Heyting, Ed. L. E. J. Brouwer: Collected Works 1: Philosophy and Foundations of Mathematics. Amsterdam: North Holland / New York: American Elsevier (1975): pp. 107-111.

Go31 Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York.

Hi27 David Hilbert. 1927. The Foundations of Mathematics. In The Emergence of Logical Empiricism. 1996. Garland Publishing Inc.

Hi30 David Hilbert. 1930. Die Grundlegung der elementaren Zahlenlehre. Mathematische Annalen. Vol. 104 (1930), pp. 485-494.

Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand. pp.145-146.

Ro36 J. Barkley Rosser. 1936. Extensions of some Theorems of Gödel and Church. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from The Journal of Symbolic Logic. Vol.1. pp.87-91.

Sc67 Joseph R. Schoenfield. 1967. Mathematical Logic. Reprinted 2001. A. K. Peters Ltd., Massachusetts.

Tu36 Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

Wi56 Ludwig Wittgenstein. 1956. Remarks on the Foundations of Mathematics. Edited by G. H. von Wright and R. Rhees. Translated by G. E. M. Anscombe. Basil Blackwell, Oxford.

Notes

Return to *: Edited and transcribed from this 2010 preprint. Some of its pedantic conclusions regarding the `soundness’ of the standard interpretation of PA (and consequences thereof) should, however, be treated as qualified by the broader philosophical perspective that treats the standard and algorithmic interpretations of PA as complementary—rather than contradictory—interpretations (as detailed in this post).

Return to 1: We show in this paper that, from a finitary perspective (such as that of this preprint) the proofs of both of Gödel’s celebrated theorems in Go31—his Theorem VI postulating the existence of an undecidable proposition in his formal Peano Arithmetic, P, and his Theorem XI postulating that the consistency of P can be expressed, but not proven, within P—hold vacuously for first order Peano Arithmetic, PA.

Return to 2: Although we have restricted ourselves in this paper to considering only PA, the arguments would—prima facie—apply equally to any first-order theory that contains sufficient Peano Arithmetic in Gödel’s sense (cf. Go31, p.28(2)), by which we mean that every primitive recursive relation is definable within the theory in the sense of Gödel’s Theorems V (Go31, p.22) and VII (Go31, p.29).

Return to 3: Following Hilbert.

Return to 4: Since we cannot, then, have that $[\neg(\forall x)\neg R(x)]$ is PA-provable and that $[\neg R(n)]$ is also PA-provable for any given numeral $[n]$.

Return to 5: This corresponds to the P-formula of his paper that Gödel defines, and refers to, only by its Gödel-number $r$ (cf. Go31, p.25, eqn.(12)).

Return to 6: Gödel refers to these P-formulas only by their Gödel-numbers $17Gen \hspace{+.5ex} r$ and $Neg(17Gen \hspace{+.5ex} r)$ respectively (cf. Go31, p.25, eqn.13).

Return to 7: i.e., for any given natural number $n$.

Return to 8: Because the arithmetical relation $R(x)$ is a Halting-type of relation (cf.Tu36, $\S 8$) that is constructively computable as true for any given natural number $n$, although it is not Turing-computable as true for any given natural number $n$ (as detailed in this post).

Return to 9: Gödel refers to it only by its Gödel-number $w$ (Go31, p.37).

Return to 10: Gödel’s broader conclusion—unchallenged so far but questionable—was that his reasoning could be validly “… carried over, word for word, to the axiom systems of set theory M and of classical mathematics A”.

Return to 11: That Gödel was open to such a possibility in 1931 is evidenced by his remark (Go31, p37) that “… it is conceivable that there might be finitary proofs which cannot be represented in P (or in M or A)”.

It is a misconception that an arithmetical statement—such as the one constructed by Kurt Gödel (1931. On formally undecidable propositions of Principia Mathematica and related systems I. In M. Davis. 1965. The Undecidable. p25)—can be intuitively true, and yet not follow formally from the axioms and rules of inference of a first-order Peano Arithmetic, $PA$.

The misconception arises because $PA$ actually admits two logical entailments, only one of which—Gödelian provability—has, so far, been formally acknowledged.

However, the other—familiar only in its avatar as the intuitive truth of a proposition under $PA$‘s standard interpretation—does, also, follow formally from the axioms and rules of inference of $PA$.

Even when this issue is sought to be addressed, the argument is indirect, and this point remains implicit.

For instance, in a critical review of Roger Penrose’s Gödelian argument, Martin Davis (1990. Is Mathematical Insight Algorithmic? Behavioural and Brain Sciences, vol. 13 (1990), pp. 659–660) argues that:

“… There is an algorithm which, given any consistent set of axioms, will output a polynomial equation $P = 0$ which in fact has no integer solutions, but such that this fact can not be deduced from the given axioms. Here then is the true but unprovable Gödel sentence on which Penrose relies and in a particularly simple form at that. Note that the sentence is provided by an algorithm. If insight is involved, it must be in convincing oneself that the given axioms are indeed consistent, since otherwise we will have no reason to believe that the Gödel sentence is true”.

Note that the first part of Gödel’s argument in Theorem VI of his 1931 paper is that, if $PA$ is consistent, then we can mechanically construct a $PA$ formula—which, syntactically, is of the form $[(\forall x)R(x)]$—such that:

(i) The formula $[(\forall x)R(x)]$, when viewed as a string of ‘meaningless’ symbols, does not follow mechanically from the axioms of $PA$ as the last of any finite sequence of $PA$-formulas, each of which is either a $PA$-axiom, or a consequence of one or more of the formulas preceding it in the sequence, by the mechanical application of the rules of inference of $PA$;

(ii) For any given numeral $[n]$—which ‘represents’ the natural number $n$ in $PA$—the formula $[R(n)]$, when viewed as a string of ‘meaningless’ symbols, does follow mechanically from the axioms of $PA$ as the last of some finite sequence of $PA$-formulas, each of which is either a $PA$-axiom, or a consequence of one or more of the formulas preceding it in the sequence, by the mechanical application of the rules of inference of $PA$.

Now, (i) is the standard definition (due to Gödel) of the meta-assertion:

(iii) The $PA$-formula $[(\forall x)R(x)]$ is formally unprovable in $PA$.

However, under standard interpretations of Alfred Tarski’s definitions of the satisfiability and truth of the formulas of a language $L$ under an interpretation $M$, the $L$-formula $[(\forall x)R(x)]$ is true in the interpretation $M$ if, and only if, the interpreted relation $R^{\prime}(x)$ is instantiationally satisfied in $M$ (i. e. for any given element of $M$ the interpreted relation can be ‘seen’ to hold in the interpretation).

If we take both $L$ and $M$ as $PA$ (as detailed in ‘Evidence-Based Interpretations of PA‘), and take satisfiability in $PA$ to mean instantiational provability in $PA$, we arrive at the formal definition of the truth of the $PA$-formula $[(\forall x)R(x)]$ in $PA$ as:

The $PA$-formula $[(\forall x)R(x)]$ is formally true in $PA$ if, and only if, the formula $[R(x)]$ is provable in $PA$ whenever we substitute a numeral $[n]$ for the variable $[x]$ in $[R(x)]$.

Hence (ii) is the standard definition (due to Tarski) of the meta-assertion:

(iv) The $PA$-formula $[(\forall x)R(x)]$ is formally true in $PA$.

So, by definition, the appropriate interpretation of Gödel’s reasoning (i) and (ii) ought to be:

(v) The $PA$-formula $[(\forall x)R(x)]$ is formally unprovable in $PA$, but formally true in $PA$.

This interpretation also meets Ludwig Wittgenstein’s (Remarks on the Foundations of Mathematics. 1978 edition. MIT Press) requirement that the concept of ‘truth’ in a language must be formally definable, and effectively verifiable, within the language.

As noted by Reuben L. Goodstein (1972. Wittgenstein’s Philosophy of Mathematics. In Ambrose, Alice, and Morris Lazerowitz (eds.), Ludwig Wittgenstein: Philosophy and Language. George Allen and Unwin. pp. 271–86):

“In the realist-formalist controversy in the philosophy of mathematics Wittgenstein’s Remarks offers a solution that is crystal clear and satisfyingly uncompromising. The true propositions of mathematics are true because they are provable in a calculus; they are deductions from axioms by formal rules and are true in virtue of valid applications of the rules of inference and owe nothing to the world outside mathematics.”

However, standard expositions of Gödel’s formal reasoning assert only that:

(vi) The $PA$-formula $[(\forall x)R(x)]$ is formally unprovable in $PA$, but intuitively true in the standard interpretation of $PA$.

They fail to highlight that, actually, (i) and (ii) are both logically entailed by the axioms and rules of inference of $PA$, and that, classically, the meta-assertion:

(vii) The $PA$-formula $[(\forall x)R(x)]$ is intuitively true in the standard interpretation of $PA$.

is both ambiguous and stronger than the meta-assertion:

(viii) The $PA$-formula $[R(x)]$ is formally true in $PA$.

The ambiguity surfaces in the presence of the Church-Turing Thesis, for (vii), then, implicitly implies that the arithmetical relation $R(x)$ is algorithmically decidable as always true in the standard interpretation of $PA$, whereas (viii) does not.

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