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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.)

Ferguson’s and Priest’s thesis

In a brief, but provocative, review of what they term as “the enduring evolution of logic” over the ages, the authors of Oxford University Press’ recently released ‘A Dictionary of Logic‘, philosophers Thomas Macaulay Ferguson and Graham Priest, take to task what they view as a Kant-influenced manner in which logic is taught as a first course in most places in the world:

“… as usually ahistorical and somewhat dogmatic. This is what logic is; just learn the rules. It is as if Frege had brought down the tablets from Mount Sinai: the result is God-given, fixed, and unquestionable.”

Ferguson and Priest conclude their review by remarking that:

“Logic provides a theory, or set of theories, about what follows from what, and why. And like any theoretical inquiry, it has evolved, and will continue to do so. It will surely produce theories of greater depth, scope, subtlety, refinement—and maybe even truth.”

However, it is not obvious whether that is prescient optimism, or a tongue-in-cheek exit line!

A nineteenth century parody of the struggle to define ‘truth’ objectively

For, if anything, the developments in logic since around 1931 has—seemingly in gross violation of the hallowed principle of Ockham’s razor, and its crude, but highly effective, modern avatar KISS—indeed produced a plethora of theories of great depth, scope, subtlety, and refinement.

These, however, seem to have more in common with the, cynical, twentieth century emphasis on subjective, unverifiable, ‘truth’, rather than with the concept of an objective, evidence-based, ‘truth’ that centuries of philosophers and mathematicians strenuously struggled to differentiate and express.

A struggle reflected so eloquently in this nineteenth century quote:

“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”

“The question is,” said Alice, “whether you can make words mean so many different things.”

“The question is,” said Humpty Dumpty, “which is to be master—that’s all.”

… Lewis Carroll (Charles L. Dodgson), ‘Through the Looking-Glass’, chapter 6, p. 205 (1934 ed.). First published in 1872.

Making sense of mathematical propositions about infinite processes

It was, indeed, an epic struggle which culminated in the nineteenth century standards of rigour successfully imposed—in no small measure by the works of Augustin-Louis Cauchy and Karl Weierstrasse—on verifiable interpretations of mathematical propositions about infinite processes involving real numbers.

A struggle, moreover, which should have culminated equally successfully in similar twentieth century standards—on verifiable interpretations of mathematical propositions containing references to infinite computations involving integers—sought to be imposed in 1936 by Alan Turing upon philosophical and mathematical discourse.

For it follows from Turing’s 1936 reasoning that where quantification is not, or cannot be, explicitly defined in formal logical terms—eg. the classical expression of the Liar paradox as ‘This sentence is a lie’—a paradox cannot per se be considered as posing serious linguistic or philosophical concerns (see, for instance, the series of four posts beginning here).

Of course—as reflected implicitly in Kurt Gödel’s seminal 1931 paper on undecidable arithmetical propositions—it would be a matter of serious concern if the word ‘This’ in the English language sentence, ‘This sentence is a lie’, could be validly viewed as implicitly implying that:

(i) there is a constructive infinite enumeration of English language sentences;

(ii) to each of which a truth-value can be constructively assigned by the rules of a two-valued logic; and,

(iii) in which ‘This’ refers uniquely to a particular sentence in the enumeration.

Gödel’s influence on Turing’s reasoning

However, Turing’s constructive perspective had the misfortune of being subverted by a knee-jerk, anti-establishment, culture that was—and apparently remains to this day—overwhelmed by Gödel’s powerful Platonic—and essentially unverifiable—mathematical and philosophical 1931 interpretation of his own construction of an arithmetical proposition that is formally unprovable, but undeniably true under any definition of ‘truth’ in any interpretation of arithmetic over the natural numbers.

Otherwise, I believe that Turing could easily have provided the necessary constructive interpretations of arithmetical truth—sought by David Hilbert for establishing the consistency of number theory finitarily—which is addressed by the following paper due to appear in the December 2016 issue of ‘Cognitive Systems Research‘:

What is logic: using Ockham’s razor

Moreover, the paper endorses the implicit orthodoxy of an Ockham’s razor influenced perspective—which Ferguson and Priest seemingly find wanting—that logic is simply a deterministic set of rules that must constructively assign the truth values of ‘truth/falsity’ to the sentences of a language.

It is a view that I expressed earlier as the key to a possible resolution of the EPR paradox in the following paper that I presented on 26’th June at the workshop on Emergent Computational Logics at UNILOG’2015, Istanbul, Turkey:

where I introduced the definition:

A finite set $\lambda$ of rules is a Logic of a formal mathematical language $\mathcal{L}$ if, and only if, $\lambda$ constructively assigns unique truth-values:

(a) Of provability/unprovability to the formulas of $\mathcal{L}$; and

(b) Of truth/falsity to the sentences of the Theory $T(\mathcal{U})$ which is defined semantically by the $\lambda$-interpretation of $\mathcal{L}$ over a structure $\mathcal{U}$.

I showed there that such a definitional rule-based approach to ‘logic’ and ‘truth’ allows us to:

$\bullet$ Equate the provable formulas of the first order Peano Arithmetic PA with the PA formulas that can be evidenced as true’ under an algorithmically computable interpretation of PA over the structure $\mathbb{N}$ of the natural numbers;

$\bullet$ Adequately represent some of the philosophically troubling abstractions of the physical sciences mathematically;

$\bullet$ Interpret such representations unambiguously; and

$\bullet$ Conclude further:

$\bullet$ First that the concept of infinity is an emergent feature of any mechanical intelligence whose true arithmetical propositions are provable in the first-order Peano Arithmetic; and

$\bullet$ Second that discovery and formulation of the laws of quantum physics lies within the algorithmically computable logic and reasoning of a mechanical intelligence whose logic is circumscribed by the first-order Peano Arithmetic.

Author’s working archives & abstracts of investigations

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

A Economist: The return of the machinery question

In a Special Report on Artificial Intelligence in its issue of 25th June 2016, ‘The return of the machinery question‘, the Economist suggests that both cosmologist Stephen Hawking and enterpreneur Elon Musk share to some degree the:

“… fear that AI poses an existential threat to humanity, because superintelligent computers might not share mankind’s goals and could turn on their creators”.

B Our irrational propensity to fear that which we are drawn to embrace

Surprising, since I suspect both would readily agree that, if anything should scare us, it is our irrational propensity to fear that which we are drawn to embrace!

And therein should lie not only our comfort, but perhaps also our salvation.

For Artificial Intelligence is constrained by rationality; Human Intelligence is not.

An Artificial Intelligence must, whether individually or collectively, create and/or destroy only rationally. Humankind can and does, both individually and collectively, create and destroy irrationally.

C Justifying irrationality

For instance, as the legatees of logicians Kurt Goedel and Alfred Tarski have amply demonstrated, a Human Intelligence can easily be led to believe that some statements of even the simplest of mathematical languages—Arithmetic—must be both ‘formally undecidable’ and ‘true’, even in the absence of any objective yardstick for determining what is ‘true’!

D Differentiating between Human reasoning and Mechanistic reasoning

An Artificial Intelligence, however, can only treat as true that which can be proven—by its rules—to be true by an objective assignment of ‘truth’ and ‘provability’ values to the propositions of the language that formally expresses its mechanical operations—Arithmetic.

The implications of the difference are not obvious; but that the difference could be significant is the thesis of this paper which is due to appear in the December 2016 issue of Cognitive Systems Research:

E Respect for evidence-based ‘truth’ could be Darwinian

More importantly, the paper demonstrates that both Human Intelligence—whose evolution is accepted as Darwinian—and Artificial Intelligence—whose evolution it is ‘feared’ may be Darwinian—share a common (Darwinian?) respect for an accountable concept of ‘truth’.

A respect that should make both Intelligences fitter to survive by recognising what philosopher Christopher Mole describes in this invitational blogpost as the:

“… importance of the rapport between an organism and its environment”

—an environment that can obviously accommodate the birth, and nurture the evolution, of both intelligences.

So, it may not be too far-fetched to conjecture that the evolution of both intelligences must also, then, share a Darwinian respect for the kind of human values—towards protecting intelligent life forms—that, no matter in how limited or flawed a guise, is visibly emerging as an inherent characteristic of a human evolution which, no matter what the cost could, albeit optimistically, be viewed as struggling to incrementally strengthen, and simultaneously integrate, individualism (fundamental particles) into nationalism (atoms) into multi-nationalism (molecules) and, possibly, into universalism (elements).

F The larger question: Should we fear an extra-terrestrial Intelligence?

From a broader perspective yet, our apprehensions about the evolution of a rampant Artificial Intelligence created by a Frankensteinian Human Intelligence should, perhaps, more rightly be addressed—as some have urged—within the larger uncertainty posed by SETI:

Is there a rational danger to humankind in actively seeking an extra-terrestrial intelligence?

I would argue that any answer would depend on how we articulate the question and that, in order to engage in a constructive and productive debate, we need to question—and reduce to a minimum—some of our most cherished mathematical and scientific beliefs and fears which cannot be communicated objectively.

Author’s working archives & abstracts of investigations

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

We investigate whether the probabilistic distribution of prime numbers can be treated as a heuristic model of quantum behaviour, since it too can be treated as a quantum phenomena, with a well-defined binomial probability function that is algorithmically computable, where the conjectured values of $\pi(n)$ differ from actual values with a binomial standard deviation, and where we define a phenomena as a quantum phenomena if, and only if, it obeys laws that can only be represented mathematically by functions that are algorithmically verifiable, but not algorithmically computable.

1. Thesis: The concept of ‘mathematical truth’ must be accountable

The thesis of this investigation is that a major philosophical challenge—which has so far inhibited a deeper understanding of the quantum behaviour reflected in the mathematical representation of some laws of nature (see, for instance, this paper by Eamonn Healey)—lies in holding to account the uncritical acceptance of propositions of a mathematical language as true under an interpretation—either axiomatically or on the basis of subjective self-evidence—without any specified methodology of accountability for objectively evidencing such acceptance.

2. The concept of ‘set-theoretical truth’ is not accountable

Since current folk lore is that all scientific truths can be expressed adequately, and communicated unambiguously, in the first order Set Theory ZF, and since the Axiom of Infinity of ZF cannot—even in principle—be objectively evidenced as true under any putative interpretation of ZF (as we argue in this post), an undesirable consequence of such an uncritical acceptance is that the distinction between the truths of mathematical propositions under interpretation which can be objectively evidenced, and those which cannot, is not evident.

3. The significance of such accountability for mathematics

The significance of such a distinction for mathematics is highlighted in this paper due to appear in the December 2016 issue of Cognitive Systems Research, where we address this challenge by considering the two finitarily accountable concepts of algorithmic verifiability and algorithmic computability (first introduced in this paper at the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, Birmingham, UK).

(i) Algorithmic verifiability

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 $\{F(1), F(2), \ldots, F(n)\}$.

(ii) Algorithmic computability

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

(iii) Algorithmic verifiability vis à vis algorithmic computability

We note that algorithmic computability implies the existence of an algorithm that can 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 decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions.

From the point of view of a finitary mathematical philosophy—which is the constraint within which an applied science ought to ideally operate—the significant difference between the two concepts could be expressed by saying that we may treat the decimal representation of a real number as corresponding to a physically measurable limit—and not only to a mathematically definable limit—if and only if such representation is definable by an algorithmically computable function (Thesis 1 on p.9 of this paper that was presented on 26th June at the workshop on Emergent Computational Logics at UNILOG’2015, 5th World Congress and School on Universal Logic, Istanbul, Turkey).

We note that although every algorithmically computable relation is algorithmically verifiable, the converse is not true.

We show in the CSR paper how such accountability helps define finitary truth assignments that differentiate human reasoning from mechanistic reasoning in arithmetic by identifying two, hitherto unsuspected, Tarskian interpretations of the first order Peano Arithmetic PA, under both of which the PA axioms interpret as finitarily true over the domain $N$ of the natural numbers, and the PA rules of inference preserve such truth finitarily over $N$.

4. The ambit of human reasoning vis à vis the ambit of mechanistic reasoning

One corresponds to the classical, non-finitary, putative standard interpretation of PA over $N$, and can be treated as circumscribing the ambit of human reasoning about ‘true’ arithmetical propositions.

The other corresponds to a finitary interpretation of PA over $N$ that circumscibes the ambit of mechanistic reasoning about ‘true’ arithmetical propositions, and establishes the long-sought for consistency of PA (see this post); which establishes PA as a mathematical language of unambiguous communication for the mathematical representation of physical phenomena.

5. The significance of such accountability for the mathematical representation of physical phenomena

The significance of such a distinction for the mathematical representation of physical phenomena is highlighted in this paper that was presented on 26th June at the workshop on Emergent Computational Logics at UNILOG’2015, 5th World Congress and School on Universal Logic, Istanbul, Turkey, where we showed how some of the seemingly paradoxical elements of quantum mechanics may resolve if we define:

Quantum phenomena: A phenomena is a quantum phenomena if, and only if, it obeys laws that can only be represented mathematically by functions that are algorithmically verifiable but not algorithmically computable.

6. The mathematical representation of quantum phenomena that is determinate but not predictable

By considering the properties of Gödel’s $\beta$ function (see $\S$4.1 on p.8 of this preprint)—which allows us to strongly represent any non-terminating sequence of natural numbers by an arithmetical function—it would follow that, since any projection of the future values of a quantum-phenomena-associated, algorithmically verifiable, function is consistent with an infinity of algorithmically computable functions, all of whose past values are identical to the algorithmically verifiable past values of the function, the phenomena itself would be essentially unpredicatable if it cannot be represented by an algorithmically computable function.

However, since the algorithmic verifiability of any quantum phenomena shows that it is mathematically determinate, it follows that the physical phenomena itself must observe determinate laws.

7. Such representation does not need to admit multiverses

Hence (contrary to any interpretation that admits unverifiable multiverses) only one algorithmically computable extension of the function is consistent with the law determining the behaviour of the phenomena, and each possible extension must therefore be associated with a probability that the next observation of the phenomena is described by that particular extension.

8. Is the probability of the future behaviour of quantum phenomena definable by an algorithmically computable function?

The question arises: Although we cannot represent quantum phenomena explicitly by an algorithmically computable function, does the phenomena lend itself to an algorithmically computable probability of its future behaviour in the above sense?

9. Can primes yield a heuristic model of quantum behaviour?

We now show that the distribution of prime numbers denoted by the arithmetical prime counting function $\pi(n)$ is a quantum phenomena in the above sense, with a well-defined probability function that is algorithmically computable.

10. Two prime probabilities

We consider the two probabilities:

(i) The probability $P(a)$ of selecting a number that has the property of being prime from a given set $S$ of numbers;

Example 1: I have a bag containing $100$ numbers in which there are twice as many composites as primes. What is the probability that the first number you blindly pick from it is a prime. This is the basis for setting odds in games such as roulette.

(ii) The probability $P(b)$ of determining a proper factor of a given number $n$.

Example 2: I give you a $5$-digit combination lock along with a $10$-digit number $n$. The lock only opens if you set the combination to a proper factor of $n$ which is greater than $1$. What is the probability that the first combination you try will open the lock. This is the basis for RSA encryption, which provides the cryptosystem used by many banks for securing their communications.

11. The probability of a randomly chosen number from the set of natural numbers is not definable

Clearly the probability $P(a)$ of selecting a number that has the property of being prime from a given set $S$ of numbers is definable if the precise proportion of primes to non-primes in $S$ is definable.

However if $S$ is the set $N$ of all integers, and we cannot define a precise ratio of primes to composites in $N$, but only an order of magnitude such as $O(\frac{1}{log_{_{e}}n})$, then equally obviously $P(a) = P(n\ is\ a\ prime)$ cannot be defined in $N$ (see Chapter 2, p.9, Theorem 2.1, here).

12. The prime divisors of a natural number are independent

Now, the following paper proves $P(b) = \frac{1}{\pi(\sqrt{n})}$, since it shows that whether or not a prime $p$ divides a given integer $n$ is independent of whether or not a prime $q \neq p$ divides $n$:

Why Integer Factorising cannot be polynomial time

We thus have that $\pi(n) \approx n.\prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}})$, with a binomial standard deviation.

Hence, even though we cannot define the probability $P(n\ is\ a\ prime)$ of selecting a number from the set $N$ of all natural numbers that has the property of being prime, $\prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}})$ can be treated as the putative non-heuristic probability that a given $n$ is a prime.

13. The distribution of primes is a quantum phenomena

The distribution of primes is thus determinate but unpredictable, since it is representable by the algorithmically verifiable but not algorithmically computable arithmetical number-theoretic function $Pr(n) = p_{_{n}}$, where $p_{_{n}}$ is the $n$‘th prime.

The Prime Number Generating Theorem and the Trim and Compact algorithms detailed in this 1964 investigation illustrate why the arithmetical number-theoretic function $Pr(n)$ is algorithmically verifiable but not algorithmically computable (see also this Wikipedia proof that no non-constant polynomial function $Pr(n)$ with integer coefficients exists that evaluates to a prime number for all integers $n$.).

Moreover, although the distribution of primes is a quantum phenomena with probabilty $\prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}})$, it is easily seen (see Figs. 7-11 on pp.23-26 of this preprint) that the generation of the primes is algorithmically computable.

14. Why the universe may be algorithmically computable

By analogy, this suggests that although the measurable values of some individual properties of particles in the universe over time may represent a quantum phenomena, the universe itself may be algorithmically computable if the laws governing the generation of all the particles in the universe over time are algorithmically computable.

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

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

A finitary arithmetical perspective on the forcing of non-standard models onto PA

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

In the previous post we formally argued that the first order Peano Arithmetic PA is categorical from a finitary perspective ($\S 5$, Corollary 1).

We now argue that conventional wisdom which holds PA as essentially incomplete—and thus precludes categoricity—may appeal to finitarily fragile arguments (as mentioned in this earlier post and in this preprint, now reproduced below) for the existence of non-standard models of the first-order Peano Arithmetic PA.

Such wisdom ought, therefore, to be treated foundationally as equally fragile from a post-computationalist arithmetical perspective within classical logic, rather than accepted as foundationally sound relative to an ante-computationalist perspective of set theory.

Post-computationalist doctrine

“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 …” (cf. Mu91).

$\S 1$ Introduction

Once we accept as logically sound the set-theoretically based meta-argument [1] that the first-order Peano Arithmetic PA [2] can be forced—by an ante-computat- ionalist interpretation of the Compactness Theorem—into admitting non-standard models which contain an infinite’ integer, then the set-theoretical properties [3] of the algebraic and arithmetical structures of such putative models should perhaps follow without serious foundational reservation.

Compactness Theorem: If every finite subset of a set of sentences has a model, then the whole set has a model (BBJ03. p.147).

However we shall argue that, from an arithmetical perspective, we can only conclude by the Compactness Theorem that if $Th(\mathbb{N})$ is the $\mathcal{L}_{A}$-theory of the standard model (interpretation) (Ka91, p.10-11), then we may consistently add to it the following as an additional—not necessarily independent—axiom:

$(\exists y)(y > x)$.

Moreover, we shall argue that even though $(\exists y)(y > x)$ is algorithmically computable (Definition 2) as always true in the standard model (whence all of its instances are in $Th(\mathbb{N})$) we cannot conclude by the Compactness Theorem that:

$\cup_{k \in \mathbb{N}}\{Th(\mathbb{N}) \cup \{c > \underline{n}\ |\ n < k\}\}$

is consistent and has a model $M_{c}$ which contains an infinite’ integer [4].

Reason: We shall argue that the condition $k \in \mathbb{N}$ in the above definition of $\cup_{k \in \mathbb{N}}T_{k}$ requires, first of all, that we must be able to extend $Th(\mathbb{N})$ by the addition of a relativised’ axiom (cf. Fe92; Me64, p.192) such as:

$(\exists y)((x \in \mathbb{N}) \rightarrow (y > x))$,

from which we may conclude the existence of some $c$ such that:

$M_{c} \models c>\underline{n}$ for all $n \in \mathbb{N}$.

However, we shall further show that even this would not yield a model for $Th(\mathbb{N})$ since, by Theorem 1, we cannot introduce a completed’ infinity such as $\mathbb{N}$ into either $Th(\mathbb{N})$ or any model of $Th(\mathbb{N})$!

$\S 1.1$ A post-computationalist doctrine

More generally we shall argue that—if our interest is in the arithmetical properties of models of PA—then we first need to make explicit any appeal to non-constructive considerations such as Aristotle’s particularisation (Definition 3).

We shall then argue that, even from a classical perspective, there are serious foundational, post-computationalist, reservations to accepting that a consistent PA can be forced by the Compactness Theorem into admitting non-standard models which contain elements other than the natural numbers.

Reason: Any arithmetical application of the Compactness Theorem to PA can neither ignore currently accepted post-computationalist doctrines of objectivity—nor contradict the constructive assignments of satisfaction and truth to the atomic formulas of PA (therefore to the compound formulas under Tarski’s inductive definitions) in terms of either algorithmical verifiability or algorithmic computability (An12, $\S 3$).

The significance of this doctrine [5] is that it helps highlight how the algorithmically verifiable (Definition 1) formulas of PA define the classical non-finitary standard interpretation of PA [6] (to which standard arguments for the existence of non-standard models of PA critically appeal).

Accordingly, we shall show that standard arguments which appeal to the ante-computationalist interpretation of the Compactness Theorem—for forcing non-standard models of PA [7] which contain an infinite’ integer—cannot admit constructive assignments of satisfaction and truth [8] (in terms of algorithmical verifiability) to the atomic formulas of their putative extension of PA.

We shall conclude that such arguments therefore questionably postulate by axiomatic fiat that which they seek to prove’!

$\S 1.2$ Standard arguments for non-standard models of PA

In this limited investigation we shall consider only the following three standard arguments for the existence of non-standard models of the first-order Peano Arithmetic PA:

(i) If PA is consistent, then we obtain a non-standard model for PA which contains an infinite’ integer by applying the Compactness Theorem to the union of the set of formulas that are satisfied or true in the classical standard’ model of PA [9] and the countable set of all PA-formulas of the form $[c_{n} = S(c_{n+1})]$.

(ii) If PA is consistent, then we obtain a non-standard model for PA which contains an infinite’ integer by adding a constant $c$ to the language of PA and applying the Compactness Theorem to the theory P$\cup\{c > \underline{n}: \underline{n}\ =\ \underline{0},\ \underline{1},\ \underline{2},\ \ldots\}$.

(iii) If PA is consistent, then we obtain a non-standard model for PA which contains an infinite’ integer by adding the PA formula $[\neg (\forall x)R(x)]$ as an axiom to PA, where $[(\forall x)R(x)]$ is a Gödelian formula [10] that is unprovable in PA, even though $[R(n)$] is provable in PA for any given PA numeral $[n]$ (Go31, p.25(1)).

We shall first argue that (i) and (ii)—which appeal to Thoralf Skolem’s ante-computationalist reasoning (in Sk34) for the existence of a non-standard model of PA—should be treated as foundationally fragile from a finitary, post-computationalist perspective within classical logic [11].

We shall then argue that although (iii)—which appeals to Kurt Gödel’s (also ante-computationalist) reasoning (Go31) for the existence of a non-standard model of PA—does yield a model other than the classical standard’ model of PA, we cannot conclude by even classical (albeit post-computationalist) reasoning that the domain is other than the domain $N$ of the natural numbers unless we make the non-constructive—and logically fragile—extraneous assumption that a consistent PA is necessarily $\omega$-consistent.

($\omega$-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 $[a]$.

$\S 2$ Algorithmically verifiable formulas and algorithmically computable formulas

We begin by distinguishing between:

Definition 1: An atomic formula $[F(x)]$ [12] of PA is algorithmically verifiable under an interpretation if, and only if, for any given numeral $[n]$, there is an algorithm $AL_{(F,\ n)}$ which can provide objective evidence (Mu91) for deciding the truth value of each formula in the finite sequence of PA formulas $\{[F(1)], [F(2)], \ldots, [F(n)]\}$ under the interpretation.

The concept is well-defined in the sense 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 (An12).

We note further that the formulas of the first order Peano Arithmetic PA are decidable under the standard interpretation of PA over the domain $\mathbb{N}$ of the natural numbers if, and only if, they are algorithmically verifiable under the interpretation [13].

Definition 2: An atomic formula $[F(x)]$ of PA is algorithmically computable under an interpretation if, and only if, there is an algorithm $AL_{F}$ that can provide objective evidence for deciding the truth value of each formula in the denumerable sequence of PA formulas $\{[F(1)], [F(2)], \ldots\}$ under the interpretation.

This concept too is well-defined in the sense 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 (An12).

We note further that the PA-formulas are decidable under an algorithmic interpretation of PA over $\mathbb{N}$ if, and only if, they are algorithmically computable under the interpretation [14].

Although we shall not appeal to the following in this paper, we note in passing that the foundational significance [15] of the distinction lies in the argument that:

Lemma 1: There are algorithmically verifiable number theoretical functions which are not algorithmically computable. [16]

Proof: Let $r(n)$ denote the $n^{th}$ digit in the decimal expansion $\sum_{n=1}^{\infty}r(n).10^{-n}$ of a putatively given real number $\mathbb{R}$ in the interval $0 < \mathbb{R} \leq 1$. By the definition of a real number as the limit of a Cauchy sequence of rationals, it follows that $r(n)$ is an algorithmically verifiable number-theoretic function. Since every algorithmically computable real is countable (Tu36), Cantor’s diagonal argument (Kl52, pp.6-8) shows that there are real numbers that are not algorithmically computable. The Lemma follows. $\Box$

$\S 3$ Making non-finitary assumptions explicit

We next make explicit—and briefly review—a tacitly held fundamental tenet of classical logic which is unrestrictedly adopted as intuitively obvious by standard literature [17] that seeks to build upon the formal first-order predicate calculus FOL:

Definition 3: (Aristotle’s particularisation) This holds that from an 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 (HA28, pp.58-59) that:

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

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

$\S 3.1$ The significance of Aristotle’s particularisation for the first-order predicate calculus

Now we note that in a formal language the formula $[(\exists x)P(x)]$‘ is an abbreviation for the formula $[\neg(\forall x)\neg P(x)]$‘; and that the commonly accepted interpretation of this formula tacitly appeals to Aristotlean particularisation.

However, as L. E. J. Brouwer had noted in his seminal 1908 paper on the unreliability of logical principles (Br08), 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.

$\S 3.2$ 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 [18] as an explicit assumption in his formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions (Go31, p.23 and p.28).

Gödel explained at some length [19] 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).

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 [20].

$\S 4$ The ambiguity in admitting an infinite’ constant

We begin our consideration of standard arguments for the existence of non-standard models of PA which contain an infinite’ integer by first highlighting and eliminating an ambiguity in the argument as it is usually found in standard texts [21]:

Corollary. There is a non-standard model of P with domain the natural numbers in which the denotation of every nonlogical symbol is an arithmetical relation or function.

Proof. As in the proof of the existence of nonstandard models of arithmetic, add a constant $\infty$ to the language of arithmetic and apply the Compactness Theorem to the theory

P$\cup\{\infty \neq$ n: $n\ =\ 0,\ 1,\ 2,\ \ldots\}$

to conclude that it has a model (necessarily infinite, since all models of P are). The denotations of $\infty$ in any such model will be a non-standard element, guaranteeing that the model is non-standard. Then apply the arithmetical Löwenheim-Skolem theorem to conclude that the model may be taken to have domain the natural numbers, and the denotations of all nonlogical symbols arithmetical.”

… BBJ03, p.306, Corollary 25.3.

$\S 4.1$ We cannot force PA to admit a transfinite ordinal

The ambiguity lies in a possible interpretation of the symbol $\infty$ as a completed’ infinity (such as Cantor’s first transfinite ordinal $\omega$) in the context of non-standard models of PA. To eliminate this possibility we establish trivially that, and briefly examine why:

Theorem 1: No model of PA can admit a transfinite ordinal under the standard interpretation of the classical logic FOL[22].

Proof: Let [$G(x)$] denote the PA-formula:

$[x=0 \vee \neg(\forall y)\neg(x=Sy)]$

Since Aristotle’s particularisation is tacitly assumed under the standard interpretation of FOL, this translates in every model of PA, as:

If $x$ denotes an element in the domain of a model of PA, then either $x$ is 0, or $x$ is a successor’.

Further, in every model of PA, if $G(x)$ denotes the interpretation of [$G(x)$]:

(a) $G(0)$ is true;

(b) If $G(x)$ is true, then $G(Sx)$ is true.

Hence, by Gödel’s completeness theorem:

(c) PA proves $[G(0)]$;

(d) PA proves $[G(x) \rightarrow G(Sx)]$.

Gödel’s Completeness Theorem: In any first-order predicate calculus, the theorems are precisely the logically valid well-formed formulas (i. e. those that are true in every model of the calculus).

Further, by Generalisation:

(e) PA proves $[(\forall x)(G(x) \rightarrow G(Sx))]$;

Generalisation in PA: [$(\forall x)A$] follows from [$A$].

Hence, by Induction:

(f) $[(\forall x)G(x)]$ is provable in PA.

Induction Axiom Schema of PA: For any formula [$F(x)$] of PA:

[$F(0) \rightarrow ((\forall x)(F(x) \rightarrow F(Sx)) \rightarrow (\forall x)F(x))$]

In other words, except 0, every element in the domain of any model of PA is a successor’. Further, the standard PA axioms ensure that $x$ can only be a successor’ of a unique element in any model of PA.

Since Cantor’s first limit ordinal $\omega$ is not the successor’ of any ordinal in the sense required by the PA axioms, and since there are no infinitely descending sequences of ordinals (cf. Me64, p.261) in a model—if any—of a first order set theory such as ZF, the theorem follows. $\Box$

$\S 4.2$ Why we cannot force PA to admit a transfinite ordinal

Theorem 1 reflects the fact that we can define the usual order relation $<$‘ in PA so that every instance of the PA Axiom Schema of Finite Induction, such as, say:

(i) [$F(0) \rightarrow ((\forall x)(F(x) \rightarrow F(Sx)) \rightarrow (\forall x)F(x))$]

yields the weaker PA theorem:

(ii) [$F(0) \rightarrow ((\forall x) ((\forall y)(y < x \rightarrow F(y)) \rightarrow F(x)) \rightarrow (\forall x)F(x))$]

Now, if we interpret PA without relativisation in ZF [23]— i.e., numerals as finite ordinals, [$Sx$] as [$x \cup \left \{ x \right \}$], etc.— then (ii) always translates in ZF as a theorem:

(iii) [$F(0) \rightarrow ((\forall x)((\forall y)(y \in x \rightarrow F(y)) \rightarrow F(x)) \rightarrow (\forall x)F(x))$]

However, (i) does not always translate similarly as a ZF-theorem, since the following is not necessarily provable in ZF:

(iv) [$F(0) \rightarrow ((\forall x)(F(x) \rightarrow F(x \cup \left \{x\right \})) \rightarrow (\forall x)F(x))$]

Example: Define [$F(x)$] as [$x \in \omega$]’.

We conclude that, whereas the language of ZF admits as a constant the first limit ordinal $\omega$ which would interpret in any putative model of ZF as the (completed’ infinite) set $\omega$ of all finite ordinals:

Corollary 1: The language of PA admits of no constant that interprets in any model of PA as the set $N$ of all natural numbers.

We note that it is the non-logical Axiom Schema of Finite Induction of PA which does not allow us to introduce—contrary to what is suggested by standard texts [24]—an actual’ (or completed’) infinity disguised as an arbitrary constant (usually denoted by $c$ or $\infty$) into either the language, or a putative model, of PA [25].

$\S 5$ Forcing PA to admit denumerable descending dense sequences

The significance of Theorem 1 is seen in the next two arguments, which attempt to implicitly bypass the Theorem’s constraint by appeal to the Compactness Theorem for forcing a non-standard model onto PA [26].

However, we argue in both cases that applying the Compactness Theorem constructively—even from a classical perspective—does not logically yield a non-standard model for PA with an infinite’ integer as claimed [27].

$\S 5.1$ An argument for a non-standard model of PA

The first is the argument (Ln08, p.7) that we can define a non-standard model of PA with an infinite descending chain of successors, where the only non-successor is the null element $0$:

1. Let $<$$N$ (the set of natural numbers); $=$ (equality); $S$ (the successor function); $+$ (the addition function); $\ast$ (the product function); $0$ (the null element)$>$ be the structure that serves to define a model of PA, say $N$.

2. Let T[$N$] be the set of PA-formulas that are satisfied or true in $N$.

3. The PA-provable formulas form a subset of T[$N$].

4. Let $\Gamma$ be the countable set of all PA-formulas of the form $[c_{n} = Sc_{n+1}]$, where the index $n$ is a natural number.

5. Let T be the union of $\Gamma$ and T[$N$].

6. T[$N$] plus any finite set of members of $\Gamma$ has a model, e.g., $N$ itself, since $N$ is a model of any finite descending chain of successors.

7. Consequently, by Compactness, T has a model; call it $M$.

8. $M$ has an infinite descending sequence with respect to $S$ because it is a model of $\Gamma$.

9. Since PA is a subset of T, $M$ is a non-standard model of PA.

$\S 5.2$ Why the argument in $\S 5.1$ is logically fragile

However if—as claimed in $\S 5.1(6)$ above—$N$ is a model of T[$N$] plus any finite set of members of $\Gamma$, and the PA term $[c_{n}]$ is well-defined for any given natural number $n$, then:

$\bullet$ All PA-formulas of the form $[c_{n} = Sc_{n+1}]$ are PA-provable,

$\bullet$ $\Gamma$ is a proper sub-set of the PA-provable formulas, and

$\bullet$ T is identically T[$N$].

Reason: The argument cannot be that some PA-formula of the form $[c_{n} = Sc_{n+1}]$ is true in $N$, but not PA-provable, as this would imply that if PA is consistent then PA+$[\neg (c_{n} = Sc_{n+1})]$ has a model other than $N$; in other words, it would presume that which is sought to be proved, namely that PA has a non-standard model [28]!

Consequently, the postulated model $M$ of T in $\S 5.1(7)$ by Compactness’ is the model $N$ that defines T[$N$]. However, $N$ has no infinite descending sequence with respect to $S$, even though it is a model of $\Gamma$.

Hence the argument does not establish the existence of a non-standard model of PA with an infinite descending sequence with respect to the successor function $S$.

$\S 5.3$ A formal argument for a non-standard model of PA

The second is the more formal argument [29]:

“Let $Th(\mathbb{N})$ denote the complete $\mathcal{L}_{A}$-theory of the standard model, i.e. $Th(\mathbb{N})$ is the collection of all true $\mathcal{L}_{A}$-sentences. For each $n \in \mathbb{N}$ we let $\underline{n}$ be the closed term $(\ldots(((1+1)+1)+ \ldots +1))) (n\ 1s)$ of $\mathcal{L}_{A}$; $\underline{0}$ is just the constant symbol $0$. We now expand our language $\mathcal{L}_{A}$ by adding to it a new constant symbol $c$, obtaining the new language $\mathcal{L}_{c}$, and consider the following $\mathcal{L}_{c}$-theory with axioms

$\rho$ (for each $\rho \in Th(\mathbb{N})$)

and

$c>\underline{n}$ (for each $n \in \mathbb{N}$)

This theory is consistent, for each finite fragment of it is contained in

$T_{k} = Th(\mathbb{N}) \cup \{c > \underline{n}\ |\ n < k\}$

for some $k \in \mathbb{N}$, and clearly the $\mathcal{L}_{c}$-structure $(\mathbb{N},\ k)$ with domain $\mathbb{N},\ 0,\ 1,\ +,\ \cdot$ and $<$ interpreted naturally, and $c$ interpreted by the integer $k$, satisfies $T_{k}$. Thus by the compactness theore $\cup_{k \in \mathbb{N}}T_{k}$ is consistent and has a model $M_{c}$. The first thing to note about $M_{c}$ is that

$M_{c} \models c>\underline{n}$

for all $n \in \mathbb{N}$, and hence it contains an infinite’ integer.”

$\S 5.4$ Why the argument in $\S 5.3$ too is logically fragile

We note again that, from an arithmetical perspective, any application of the Compactness Theorem to PA cannot ignore currently accepted computationalist doctrines of objectivity (cf. Mu91) and contradict the constructive assignment of satisfaction and truth to the atomic formulas of PA (therefore to the compound formulas under Tarski’s inductive definitions) in terms of either algorithmical verifiability or algorithmic computability (An12, $\S 3$).

Accordingly, from an arithmetical perspective we can only conclude by the Compactness Theorem that if $Th(\mathbb{N})$ is the $\mathcal{L}_{A}$-theory of the standard model (interpretation), then we may consistently add to it the following as an additional—not necessarily independent—axiom:

$(\exists y)(y > x)$.

Moreover, even though $(\exists y)(y > x)$ is algorithmically computable as always true in the standard model—whence all instances of it are also therefore in $Th(\mathbb{N})$—we cannot conclude by the Compactness Theorem that $\cup_{k \in \mathbb{N}}T_{k}$ is consistent and has a model $M_{c}$ which contains an infinite’ integer.

Reason: The condition $k \in \mathbb{N}$‘ in $\cup_{k \in \mathbb{N}}T_{k}$ requires, first of all, that we must be able to extend $Th(\mathbb{N})$ by the addition of a relativised’ axiom (cf. Fe92; Me64, p.192) such as:

$(\exists y)((x \in \mathbb{N}) \rightarrow (y > x))$,

from which we may conclude the existence of some $c$ such that:

$M_{c} \models c>\underline{n}$

for all $n \in \mathbb{N}$.

However, even this would not yield a model for $Th(\mathbb{N})$ since, by Theorem 1, we cannot introduce a completed’ infinity such as $\mathbb{N}$ into any model of $Th(\mathbb{N})$!

As the argument stands, it seeks to violate finitarity by adding a new constant $c$ to the language $\mathcal{L}_{A}$ of PA that is not definable in $\mathcal{L}_{A}$ and, ipso facto, adding an atomic formula $[c=x]$ to PA whose satisfaction under any interpretation of PA is not algorithmically verifiable!

Since the atomic formulas of PA are algorithmically verifiable under the standard interpretation (An12, Corollary 2), the above conclusion too postulates that which it seeks to prove!

Moreover, the postulation would be false if $Th(\mathbb{N})$ were categorical.

Since $Th(\mathbb{N})$ must have a non-standard model if it is not categorical, we consider next whether we may conclude from Gödel’s incompleteness argument (in Go31) that any such model can have an infinite’ integer.

$\S 6$ Gödel’s argument for a non-standard model of PA

We begin by considering the Gödelian formula $[(\forall x)R(x)]$ [30] which is unprovable in PA if PA is consistent, even though the formula $[R(n)]$ is provable in a consistent PA for any given PA numeral $[n]$.

Now, it follows from Gödel’s reasoning [31] that:

Theorem 2: If PA is consistent, then we may add the PA formula $[\neg (\forall x)R(x)]$ as an axiom to PA without inviting inconsistency.

Theorem 3: If PA is $\omega$-consistent, then we may add the PA formula $[(\forall x)R(x)]$ as an axiom to PA without inviting inconsistency.

Gödel concluded from this that:

Corollary 2: If PA is $\omega$-consistent, then there are at least two distinctly different models of PA. $\Box$

If we assume that a consistent PA is necessarily $\omega$-consistent, then it follows that one of the two putative models postulated by Corollary 2 must contain elements other than the natural numbers.

We conclude that Gödel’s justification for the assumption that non-standard models of PA containing elements other than the natural numbers are logically feasible lies in his non-constructive—and logically fragile—assumption that a consistent PA is necessarily $\omega$-consistent.

$\S 6.1$ Why Gödel’s assumption is logically fragile

Now, whereas Gödel’s proof of Corollary 2 appeals to the non-constructive Aristotle’s particularisation, a constructive proof of the Corollary follows trivially from evidence-based interpretations of PA (An12).

Reason: Tarski’s inductive definitions allow us to provide finitary satisfaction and truth certificates to all atomic (and ipso facto to all compound) formulas of PA over the domain $N$ of the natural numbers in two essentially different ways:

(1) In terms of algorithmic verifiabilty (An12, $\S 4.2$); and

(2) In terms of algorithmic computability (An12, $\S 4.3$).

That there can be even one, let alone two, logically sound and finitary assignments of satisfaction and truth certificates to both the atomic and compound formulas of PA was hitherto unsuspected!

Moreover, neither the putative algorithmically verifiable’ model, nor the algorithmically computable’ model, of PA defined by these finitary satisfaction and truth assignments contains elements other than the natural numbers.

(a) Any algorithmically verifiable model of PA is necessarily over $\mathbb{N}$

For instance if, in the first case, we assume that the algorithmically verifiable atomic formulas of PA determine an algorithmically verifiable model of PA over the domain $\mathbb{N}$ of the PA numerals, then such a putative model would be isomorphic to the standard model of PA over the domain $N$ of the natural numbers (An12, $\S 4.2$ & $\S 5$, Corollary 2).

However, such a putative model of PA over $\mathbb{N}$ would not be finitary since, if the formula $[(\forall x) F(x)]$ were to interpret as true in it, then we could only conclude that, for any numeral $[n]$, there is an algorithm which will finitarily certify the formula $[F(n)]$ as true under an algorithmically verifiable interpretation in $\mathbb{N}$.

We could not conclude that there is a single algorithm which, for any numeral $[n]$, will finitarily certify the formula $[F(n)]$ as true under the algorithmically verifiable interpretation in $\mathbb{N}$.

Consequently, the PA Axiom Schema of Finite Induction would not interpret as true finitarily under the algorithmically verifiable interpretation of PA over the domain $\mathbb{N}$ of the PA numerals.

Thus the algorithmically verifiable interpretation of PA would not define a finitary model of PA.

However, if we were to assume that the algorithmically verifiable interpretation of PA defines a non-finitary model of PA, then it would follow that:

$\bullet$ PA is necessarily $\omega$-consistent;

$\bullet$ Aristotle’s particularisation holds over $N$; and

$\bullet$ The standard’ interpretation of PA also defines a non-finitary model of PA over $N$.

(b) The algorithmically computable interpretation of PA is over $\mathbb{N}$

The second case is where the algorithmically computable atomic formulas of PA determine an algorithmically computable model of PA over the domain $N$ of the natural numbers (An12, $\S 4.3$ & $\S 5.2$).

The algorithmically computable model of PA is finitary since we can show that, if the formula $[(\forall x) F(x)]$ interprets as true under it, then we may always conclude that there is a single algorithm which, for any numeral $[n]$, will finitarily certify the formula $[F(n)]$ as true in $N$ under the algorithmically computable interpretation.

Consequently we can show that all the PA axioms—including the Axiom Schema of Finite Induction—interpret finitarily as true in $N$ under the algorithmically computable interpretation of PA, and the PA Rules of Inference preserve such truth finitarily (An12, $\S 5.2$ Theorem 4).

Thus the algorithmically computable interpretation of PA defines a finitary model of PA from which we may conclude that:

$\bullet$ PA is consistent (An12, $\S 5.3$, Theorem 6).

$\S 6.2$ Why we cannot conclude that PA is necessarily $\omega$-consistent

By the way the above finitary interpretation (b) is defined under Tarski’s inductive definitions (An12, $\S 4.3$), if a PA-formula $[F]$ interprets as true in the corresponding finitary model of PA, then there is an algorithm that provides a certificate for such truth for $[F]$ in $N$; whilst if $[F]$ interprets as false in the above finitary model of PA, then there is no algorithm that can provide such a truth certificate for $[F]$ in $N$ (An12, $\S 2$).

Now, if there is no algorithm that can provide such a truth certificate for the Gödelian formula $[R(x)]$ in $N$, then we would have by definition first that the PA formula $[\neg(\forall x)R(x)]$ is true in the model, and second by Gödel’s reasoning that the formula $[R(n)]$ is true in the model for any given numeral $[n]$. Hence Aristotle’s particularisation would not hold in the model.

However, by definition if PA were $\omega$-consistent then Aristotle’s particularisation must necessarily hold in every model of PA.

It follows that unless we can establish that there is some algorithm which can provide such a truth certificate for the Gödelian formula $[R(x)]$ in $N$, we cannot make the unqualified assumption—as Gödel appears to do—that a consistent PA is necessarily $\omega$-consistent.

Conclusion

We have argued that standard arguments for the existence of non-standard models of the first-order Peano Arithmetic PA with domains other than the domain $N$ of the natural numbers should be treated as logically fragile even from within classical logic. In particular we have argued that although Gödel’s argument for the existence of a non-standard model of PA does yield a model of PA other than the classical non-finitary standard’ model, we cannot conclude from it that the domain is other than the domain $N$ of the natural numbers unless we make the non-constructive—and logically fragile—assumption that a consistent PA is necessarily $\omega$-consistent.

References

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

Be59 Evert W. Beth. 1959. The Foundations of Mathematics. In Studies in Logic and the Foundations of Mathematics. Edited by L. E. J. Brouwer, E. W. Beth, A. Heyting. 1959. North Holland Publishing Company, Amsterdam.

BF58 Paul Bernays and Abraham A. Fraenkel. 1958. Axiomatic Set Theory In Studies in Logic and the Foundations of Mathematics. Edited by L. E. J. Brouwer, E. W. Beth, A. Heyting. 1959. North Holland Publishing Company, Amsterdam.

Bo00 Andrey I. Bovykin. 2000. On order-types of models of arithmetic. Thesis submitted to the University of Birmingham for the degree of Ph.D. in the Faculty of Science, School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom, 13 April 2000.

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.

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.

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.

Fe92 Solomon Feferman. 1992. What rests on what: The proof-theoretic analysis of mathematics Invited lecture, 15th International Wittgenstein Symposium: Philosophy of Mathematics, held in Kirchberg/Wechsel, Austria, 16-23 August 1992.

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.

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.

HP98 Petr Hájek and Pavel Pudlák. 1998. Metamathematics of First-Order Arithmetic. Volume 3 of Perspectives in Mathematical Logic Series, ISSN 0172-6641. Springer-Verlag, Berlin, 1998.

Ka91 Richard Kaye. 1991. Models of Peano Arithmetic. Oxford Logic Guides, 15. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991.

Ka11 Richard Kaye. 2011. Tennenbaum’s theorem for models of arithmetic. In Set Theory, Arithmetic, and Foundations of Mathematics. Eds. Juliette Kennedy & Roman Kossak. Lecture Notes in Logic (No. 36). pp.66-79. Cambridge University Press, 2011. Cambridge Books Online. http://dx.doi.org/10.1017/CBO9780511910616.005.

Kl52 Stephen Cole Kleene. 1952. Introduction to Metamathematics. North Holland Publishing Company, Amsterdam.

Kn63 G. T. Kneebone. 1963. Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. D. Van Norstrand Company Limited, London.

Ko06 Roman Kossak and James H. Schmerl. 2006. The structure of models of Peano arithmetic. Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006.

Li64 A. H. Lightstone. 1964. The Axiomatic Method. Prentice Hall, NJ.

Ln08 Laureano Luna. 2008. On non-standard models of Peano Arithmetic. In The Reasoner, Vol(2)2 p7.

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

Mu91. Chetan R. Murthy. 1991. An Evaluation Semantics for Classical Proofs. Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991.

Nv64 P. S. Novikov. 1964. Elements of Mathematical Logic. Oliver & Boyd, Edinburgh and London.

Qu63 Willard Van Orman Quine. 1963. Set Theory and its Logic. Harvard University Press, Cambridge, Massachusette.

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

Ro53 J. Barkley Rosser. 1953. Logic for Mathematicians. McGraw Hill, New York.

Sh67 Joseph R. Shoenfield. 1967. Mathematical Logic. Reprinted 2001. A. K. Peters Ltd., Massachusetts.

Sk28 Thoralf A. Skolem. 1928. On Mathematical Logic. Text of a lecture delivered on 22nd October 1928 before the Norwegian Mathematical Association. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931 Harvard University Press, Cambridge, Massachusetts.

Sk34 Thoralf A. Skolem. 1934. \”{U}ber die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abz\”{a}hlbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundamenta Mathematicae, 23, 150-161. English version: Peano’s axioms and models of arithmetic. In Mathematical interpretations of formal systems. North Holland, Amsterdam, 1955, pp.1-14.

Sm92 Raymond M. Smullyan. 1992. Gödel’s Incompleteness Theorems. Oxford University Press, Inc., New York.

Su60 Patrick Suppes. 1960. Axiomatic Set Theory. Van Norstrand, Princeton.

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.

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

An12 Bhupinder Singh Anand. 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.

An13 … 2013. A suggested mathematical perspective for the $EPR$ argument. Presented on 7’th April at the workshop on Logical Quantum Structures‘ at UNILOG’2013, $4^{th}$ World Congress and School on Universal Logic, $29^{th}$ March 2013 – $7^{th}$ April 2013, Rio de Janeiro, Brazil.

Notes

Return to 1: By which we mean arguments such as in Ka91 (see pg.1), where the meta-theory is taken to be a set-theory such as ZF or ZFC, and the logical consistency of the meta-theory is not considered relevant to the argumentation.

Return to 2: For purposes of this investigation we may take this to be a first order theory such as the theory S defined in Me64, pp.102-103.

Return to 5: Some of the—hitherto unsuspected—consequences of this doctrine are detailed in An12.

Return to 6: An12, Corollary 2; non-finitary’ because the Axiom Schema of Finite Induction cannot be finitarily verified as true under the standard interpretation of PA with respect to truth’ as defined by the algorithmically verifiable formulas of PA.

Return to 8: cf. The standard non-constructive set-theoretical assignment-by-postulation (S5) of the satisfaction properties (S1) to (S8) in BBJ03, p.153, Lemma 13.1 (Satisfaction properties lemma), which appeals critically to Aristotle’s particularisation.

Return to 9: For purposes of this investigation we may take this to be an interpretation of PA as defined in Me64, p.107.

Return to 10: In his seminal 1931 paper Go31, Kurt Gödel defines, and refers to, the formula corresponding to $[R(x)]$ only by its Gödel’ number $r$ (op. cit., p.25, Eqn.(12)), and to the formula corresponding to $[(\forall x)R(x)]$ only by its Gödel’ number $17\ Gen\ r$.

Return to 11: By classical logic’ we mean the standard first-order predicate calculus FOL where we neither deny the Law of the Excludeds Middle, nor assume that the FOL is $\omega$-consistent (i.e., we do not assume that Aristotle’s particularisation must hold under any interpretation of the logic).

Return to 12: Notation: For the sake of convenience, we shall use square brackets to indicate that the expression enclosed by them is to be treated as denoting a formula of a formal theory, and not as denoting an interpretation.

Return to 13: However, as noted earlier, the Axiom Schema of Finite Induction cannot be finitarily verified as true under the standard interpretation of PA with respect to truth’ as defined by the algorithmically verifiable formulas of PA .

Return to 14: In this case however, the Axiom Schema of Finite Induction can be finitarily verified as true under the standard interpretation of PA with respect to truth’ as defined by the algorithmically computable formulas of PA (An12, Theorem 4).

Return to 15: The far reaching—hitherto unsuspected—consequences of this distinction for PA are detailed in An12.

Return to 16: We note that algorithmic computability implies the existence of an algorithm that can 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 decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions. From the point of view of a finitary mathematical philosophy, the significant difference between the two concepts could be expressed (An13) by saying that we may treat the decimal representation of a real number as corresponding to a physically measurable limit—and not only to a mathematically definable limit—if and only if such representation is definable by an algorithmically computable function.

Return to 17: 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, pp.45, 47, 52(ii), 214(fn); 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.

Return to 18: The significance of $\omega$-consistency for the formal system PA is highlighted inAn12.

Return to 21: cf. HP98, p.13, $\S 0.29$; Me64, p.112, Ex. 2.

Return to 22: For purposes of this investigation we may take this to be the first order predicate calculus $K$ as defined in Me64, p.57.

Return to 24: eg. HP98, p.13, $\S 0.29$; Ka91, p.11 & p.12, fig.1; BBJ03. p.306, Corollary 25.3; Me64, p.112, Ex. 2.

Return to 25: A possible reason why the Axiom Schema of Finite Induction does not admit non-finitary reasoning into either PA, or into any model of PA, is suggested in $\S 6.1$ below.

Return to 26: eg. Ln08, p.7; Ka91, pp.10-11, p.74 & p.75, Theorem 6.4.

Return to 27: And as suggested also by standard texts in such cases; eg. BBJ03. p.306, Corollary 25.3; Me64, p.112, Ex. 2.

Return to 28: To place this distinction in perspective, Adrien-Marie Legendre and Carl Friedrich Gauss independently conjectured in 1796 that, if $\pi (x)$ denotes the number of primes less than $x$, then $\pi (x)$ is asymptotically equivalent to $x$/In$(x)$. Between 1848/1850, Pafnuty Lvovich Chebyshev confirmed that if $\pi (x)$/($x$/In$(x)$) has a limit, then it must be 1. However, the crucial question of whether $\pi (x)$/($x$/In$(x)$) has a limit at all was answered in the affirmative using analytic methods independently by Jacques Hadamard and Charles Jean de la Vallée Poussin only in 1896, and using only elementary methods by Atle Selberg and Paul Erdös in 1949.

Return to 29: Ka91, pp.10-11; attributed as essentially Skolem’s argument in Sk34.

Return to 30: In his seminal 1931 paper Go31, Kurt Gödel defines, and refers to, the formula corresponding to $[R(x)]$ only by its Gödel’ number $r$ (op. cit., p.25, Eqn.(12)), and to the formula corresponding to $[(\forall x)R(x)]$ only by its Gödel’ number $17\ Gen\ r$.

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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Notes

Return to 1: For a brief recent review of such challenges, see Fe06, Fe08.

Return to 3: We take this to be the first-order theory S defined in Me64, p.102.

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 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 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 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 22: Essentially reflecting Brouwer’s objection to the assumption of Aristotle’s particularisation over an infinite domain.

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 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 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 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 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 42: See Hi27, p.472; also Br13, p.59; We27, p.482; Pa71, p.502-503.

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 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)”.

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George Lakoff has retired as Distinguished Professor of Cognitive Science and Linguistics at the University of California at Berkeley. He is now Director of the Center for the Neural Mind & Society (cnms.berkeley.edu).

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