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(*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 be understood from a broadly computational perspective”;

(iii) “ 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”.

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.

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**

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 is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence (cf. ibid Murthy 91) for deciding the truth/falsity of each proposition in the denumerable sequence .

(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 is algorithmically verifiable if, and only if, for any given natural number , there is an algorithm which can provide objective evidence for deciding the truth/falsity of each proposition in the finite sequence .

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:

“ 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 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.

(*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 -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* -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* (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.

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

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

** The Logical Issue**

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

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

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

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

(i) The concept of algorithmic verifiability is well-defined under the standard interpretation of PA over the structure of the natural numbers; and

(ii) The concept of algorithmic computability too is well-defined under the algorithmic interpretation of PA over the structure of the natural numbers; and

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

** Weakening the Church and Turing Theses**

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

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

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

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

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

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

** The need for explicitly distinguishing between `instantiational’ and `uniform’ methods**

**Why Church’s Thesis?**

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

**Kurt Gödel’s reservations**

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

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

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

**Alan Turing’s perspective**

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

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

**Boolos, Burgess and Jeffrey’s query**

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

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

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

However, as Boolos, Burgess and Jeffrey note quizically:

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

**Why should Chaitin’s constant be labelled `uncomputable’?**

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

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

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

** Distinguishing between algorithmic verifiability and algorithmic computability**

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

** Significance of Gödel’s -function**

We note first that in Theorem VII of his seminal 1931 paper on formally undecidable arithmetical propositions Gödel showed that, given a total number-theoretic function and any natural number , we can construct a primitive recursive function and natural numbers such that for all .

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

**Query 3:** Does Gödel’s Theorem VII admit construction of an arithmetical function such that:

(a) for any given natural number , there is an algorithm that can verify for all (hence may be said to be algorithmically verifiable if is recursive);

(b) there is no algorithm that can verify for all (so may be said to be algorithmically uncomputable)?

** Defining effective computability**

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

(i) Algorithmically verifiable;

(ii) Algorithmically computable.

under an interpretation.

We shall thus propose the definition:

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

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

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

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

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

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

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

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

** Gödel’s Theorem VII and algorithmically verifiable, but not algorithmically computable, arithmetical propositions**

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

**Definition 1:**

where denotes the remainder obtained on dividing by .

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

**Lemma 1:** For any given denumerable sequence of natural numbers, say , and any given natural number , we can construct natural numbers such that:

(i) ;

(ii) !;

(iii) for .

**Proof:** This is a standard result ^{[21]}.

Now we have the standard definition ^{[22]}:

**Definition 2:** A number-theoretic function is said to be representable in PA if, and only if, there is a PA formula with the free variables , such that, for any given natural numbers :

(i) if then PA proves: ;

(ii) PA proves: .

The function is said to be strongly representable in PA if we further have that:

(iii) PA proves:

**Interpretation of `‘:** The symbol `‘ denotes `uniqueness’ under an interpretation which assumes that Aristotle’s particularisation holds in the domain of the interpretation.

Formally, however, the PA formula:

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

.

We then have:

**Lemma 2** is strongly represented in PA by , which is defined as follows:

.

**Proof:** This is a standard result ^{[23]}.

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

**Lemma 3:** If is a recursive function defined by:

(i)

(ii)

where and are recursive functions of lower rank ^{[25]} that are represented in PA by well-formed formulas and ,

then is represented in PA by the following well-formed formula, denoted by :

**Proof:** This is a standard result ^{[26]}.

** What does “ is provable” assert under the standard interpretation of PA?**

Now, if the PA formula represents in PA the recursive function denoted by then by definition, for any given numerals , the formula is provable in PA; and true under the standard interpretation of PA.

We thus have that:

**Lemma 4:** “ is true under the standard interpretation of PA” is the assertion that:

Given any natural numbers , we can construct natural numbers —all functions of —such that:

(a) ;

(b) for all , ;

(c) ;

where , and are any recursive functions that are formally represented in PA by and respectively such that:

(i)

(ii) for all

(iii) and are recursive functions that are assumed to be of lower rank than .

**Proof:** For any given natural numbers and , if interprets as a well-defined arithmetical relation under the standard interpretation of PA, then we can define a deterministic Turing machine that can `construct’ the sequences:

and:

and give evidence to verify the assertion. ^{[27]}

We now see that:

**Theorem 1:** Under the standard interpretation of PA is algorithmically verifiable, but not algorithmically computable, as always true over .

**Proof:** It follows from Lemma 4 that:

(1) is PA-provable for any given numerals . Hence is true under the standard interpretation of PA. It then follows from the definition of in Lemma 3 that, for any given natural numbers , we can construct some pair of natural numbers —where are functions of the given natural numbers and —such that:

(a) for ;

(b) holds in .

Since is primitive recursive, defines a deterministic Turing machine that can `construct’ the denumerable sequence for any given natural numbers and such that:

(c) for .

We can thus define a deterministic Turing machine that will give evidence that the PA formula is true under the standard interpretation of PA.

Hence is algorithmically verifiable over under the standard interpretation of PA.

(2) Now, the pair of natural numbers are defined such that:

(a) for ;

(b) holds in ;

where is defined in Lemma 3 as !, and:

(c) ;

(d) is the `number’ of terms in the sequence .

Since is not definable for a denumerable sequence we cannot define a denumerable sequence such that:

(e) for all .

We cannot thus define a deterministic Turing machine that will give evidence that the PA formula interprets as true under the standard interpretation of PA for any given sequence of numerals .

Hence is not algorithmically computable over under the standard interpretation of PA.

The theorem follows.

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

The above theorem now suggests the following definition:

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

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

**References**

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**An07** Bhupinder Singh Anand. 2007. *Why we shouldn’t fault Lucas and Penrose for continuing to believe in the Gödelian argument against computationalism – II.* In *The Reasoner*, Vol(1)7 p2-3.

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

**Notes**

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

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

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

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

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

Return to 6: See Si97.

Return to 7: Go51.

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

Return to 9: Tu36, , p.139.

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

Return to 11: BBJ03, p. 37.

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

Return to 13: BBJ03, p.37.

Return to 14: Chaitin’s Halting Probability is given by , where the summation is over all self-delimiting programs that halt, and is the size in bits of the halting program ; see Ct75.

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

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

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

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

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

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

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

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

Return to 22: Me64, p.118.

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

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

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

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

Return to 27: A critical philosophical issue that we do not address here is whether the PA formula can be considered to interpret under a sound interpretation of PA as a well-defined predicate, since the denumerable sequences and is not equal to if is not equal to —are represented by denumerable, distinctly different, functions respectively. There are thus denumerable pairs for which yields the sequence .

**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, , any value of which can be computed effectively, although there is no single algorithm that can effectively compute .

“According to Turing’s Thesis, since is not Turing-computable, cannot be effectively computable. Why not? After all, although no Turing machine computes the function , we were able to compute at least its first few values, For since, as we have noted, the empty function we have . And it may seem that we can actually compute for any positive integer —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 , if it lies within the range of the variables of . 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 by definition.

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

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

Third, we may mean that an individually conceivable object exists, in an interpretation of , if it lies within the range of an interpreted variable of , where is a Platonic interpretation of 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 , say , **′**, **′****′**, …, unambiguously to the mathematical objects that are definable as the constant terms of .

If we can achieve this, we can then attempt to relate 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—, **′**, **′****′**, …,—of the standard intuitive interpretation, say , 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 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 , 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 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 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 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.

**An07b** … 2007. *Why we shouldn’t fault Lucas and Penrose for continuing to believe in the Gödelian argument against computationalism – I.* In *The Reasoner,* Vol(1)6 p3-4.

**An07c** … 2007. *Why we shouldn’t fault Lucas and Penrose for continuing to believe in the Gödelian argument against computationalism – II.* In *The Reasoner*, Vol(1)7 p2-3.

**An08** … 2008. *Can we really falsify truth by dictat?.* In *The Reasoner*, Vol(2)1 p7-8.

**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.

*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 ‘-consistency’ made explicitly by Gödel.

This assumption is actually logically equivalent to the philosophically questionable assertion that from the provability of we may conclude the existence of some numeral for which 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 to the system . 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):

on the analogy of the so-called `Liar paradox’;

on Gödel’s interpretation of his arithmetical proposition as asserting its own formal unprovability in his formal Peano Arithmetic P (Go31, pp.9-13);

on his interpretation of the reasons for the `incompleteness’ of P; and

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 -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 -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 -Rule:* If it is proved that the P-formula [] interprets as a true numerical formula for each given P-numeral [], then the P-formula 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 -rule in the limited—and more precise—sense that if a formula [] is provable in P for each given numeral [], then the formula [] must be provable in P.

Now, if we meta-assume Hilbert’s -rule for P, then it follows that, if P is consistent, then there is no P-formula [] for which, first, [] is P-provable and, second, [] is P-provable for any given P-numeral [].

Gödel defined a consistent Peano Arithmetic with the above property as additionally -consistent (Go31, pp.23-24).

**D: The significance of -consistency**

To place the significance of -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 [], which is merely an abbreviation for , interprets as true if, and only if, holds for some natural number under

*) is*

**M***sound*(cf. BBJ03, p.174).

Clearly, if such an interpretation of the existential quantifier is sound, it immediately implies that PA is necessarily -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 -consistency for his Peano Arithmetic P under the unqualified—and, as we show below, mistaken—belief that:

*PA is -consistent (Go31, p.28, footnote 48a).*

**E: Gödel: If the -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, [] ^{[5]}, such that is P-provable for any given P-numeral [] (Go31, p.25(2)), but [] 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 -consistent, then both [] and [] ^{[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 -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 -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* –*in*consistent.

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 –*in*consistent (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 [] that [ ] is PA-provable, and that [] is PA-provable for any given PA-numeral [].

Consequently, Gödel’s Theorem VI is vacuously true for PA, and it also follows that Hilbert’s -Rule is inconsistent with PA!

**G: Need: A paradigm shift in interpreting the quantifiers**

Thus Gödel’s unqualified belief that:

“*PA is -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 –*in*consistent, from which it follows that the standard interpretation * M* of PA is

*not*sound.

Hence we can no longer interpret `[] is true’ maximally under the standard interpretation of PA as:

(i) The arithmetical relation 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 of PA, we *can* interpret `[] is true’ minimally as:

(ii) The arithmetical relation 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 [] 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 -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 -consistent (as detailed in this post);

(iii) PA is –*in*consistent (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 []—defined as { (*the set of natural numbers*); (*equality*); (*the successor function*); (*the addition function*); (*the product function*); (*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 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 defines a Turing-machine TM which computes as true under if, and only if, the corresponding PA-formula [] 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 [] ^{[9]} in P and assumes that [] 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 [] 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:

() A primitive recursive relation, *Form*(), which is true if, and only if, is the Gödel-number of a formula of P;

() A primitive recursive relation, , which is true if, and only if, is the Gödel-number of a proof sequence of P whose last formula has the Gödel-number .

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 `‘, `‘, ` (There is a unique )’ occurs it is followed by a bound for . This bound serves only to assure the recursive nature of the defined concept.*

Gödel then defines a meta-mathematical proposition that is not recursive:

() A proposition, , which is true if, and only if, is true.

Thus is true if, and only if, 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 , can be represented in P by some, corresponding, arithmetical formula, say [], such that, for any natural number :

If is true, then [] is P-provable;

If is false, then [] 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 is true (i.e, is true for any given natural number), then is P-provable.

In other words, we cannot assume that, even though the recursive relation is instantiationally equivalent to a sound interpretation of the P-formula [], the number-theoretic proposition must, necessarily, be logically equivalent to the—correspondingly sound—interpretation of the P-formula [].

The reason: In recursive arithmetic, the expression `‘ is an abbreviation for the assertion:

(*) There is some (at least one) natural number such that holds.

In a formal Peano Arithmetic, however, the formula `[]’ is simply an abbreviation for `[]’, which, under a sound finitary interpretation of the Arithmetic can have the verifiable translation:

(**) The relation 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:

This translates as:

There is a natural number 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 [] of P such that “[] 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, (1931, p24(8.1)), is represented by the P-formula [], then the proposition “[] is true” is logically equivalent to (i.e., has the same meaning as) “ is true” under a sound interpretation of P.

**P: The loophole in Gödel’s presumption**

Although, (ii), for instance, does follow if “[] is true” translates as “ is Turing-computable as always true”, it does not if “[] is true” translates as “ is constructively computable as true for any given natural number , but it is not Turing-computable as true for any given natural number “.

So, if [], too, interprets as an arithmetical proposition that is constructively computable as true, but not Turing-computable as true, then the consistency of P may be provable instantiationally in P ^{[11]}.

Hence, at best, Gödel’s reasoning can only be taken to establish that the consistency of P is not provable algorithmically in P.

Gödel’s broader conclusion only follows if P purports to prove its own consistency algorithmically.

However, Gödel’s particular argument, based on his definition of *Wid*(P), does not support this claim.

**Bibliography**

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

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

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

**Hi27** David Hilbert. 1927. *The Foundations of Mathematics.* In *The Emergence of Logical Empiricism.* 1996. Garland Publishing Inc.

**Hi30** David Hilbert. 1930. *Die Grundlegung der elementaren Zahlenlehre.* Mathematische Annalen. Vol. 104 (1930), pp. 485-494.

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

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

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

**Tu36** Alan Turing. 1936. *On computable numbers, with an application to the Entscheidungsproblem.* Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

**Wi56** Ludwig Wittgenstein. 1956. *Remarks on the Foundations of Mathematics.* Edited by G. H. von Wright and R. Rhees. Translated by G. E. M. Anscombe. Basil Blackwell, Oxford.

**Notes**

Return to *: Edited and transcribed from this 2010 preprint. Some of its pedantic conclusions regarding the `soundness’ of the standard interpretation of PA (and consequences thereof) should, however, be treated as qualified by the broader philosophical perspective that treats the standard and algorithmic interpretations of PA as complementary—rather than contradictory—interpretations (as detailed in this post).

Return to 1: We show in this paper that, from a finitary perspective (such as that of this preprint) the proofs of both of Gödel’s celebrated theorems in Go31—his Theorem VI postulating the existence of an undecidable proposition in his formal Peano Arithmetic, P, and his Theorem XI postulating that the consistency of P can be expressed, but not proven, within P—hold vacuously for first order Peano Arithmetic, PA.

Return to 2: Although we have restricted ourselves in this paper to considering only PA, the arguments would—prima facie—apply equally to any first-order theory that contains sufficient Peano Arithmetic in Gödel’s sense (cf. Go31, p.28(2)), by which we mean that every primitive recursive relation is definable within the theory in the sense of Gödel’s Theorems V (Go31, p.22) and VII (Go31, p.29).

Return to 3: Following Hilbert.

Return to 4: Since we cannot, then, have that is PA-provable and that is also PA-provable for any given numeral .

Return to 5: This corresponds to the P-formula of his paper that Gödel defines, and refers to, only by its Gödel-number (cf. Go31, p.25, eqn.(12)).

Return to 6: Gödel refers to these P-formulas only by their Gödel-numbers and respectively (cf. Go31, p.25, eqn.13).

Return to 7: i.e., for any given natural number .

Return to 8: Because the arithmetical relation is a Halting-type of relation (cf.Tu36, ) that is constructively computable as true for any given natural number , although it is not Turing-computable as true for any given natural number (as detailed in this post).

Return to 9: Gödel refers to it only by its Gödel-number (Go31, p.37).

Return to 10: Gödel’s broader conclusion—unchallenged so far but questionable—was that his reasoning could be validly “… carried over, word for word, to the axiom systems of set theory M and of classical mathematics A”.

Return to 11: That Gödel was open to such a possibility in 1931 is evidenced by his remark (Go31, p37) that “… it is conceivable that there might be finitary proofs which cannot be represented in P (or in M or A)”.

It is a misconception that an arithmetical statement—such as the one constructed by Kurt Gödel (1931. *On formally undecidable propositions of Principia Mathematica and related systems I.* In M. Davis. 1965. *The Undecidable.* p25)—can be *intuitively* true, and yet not follow *formally* from the axioms and rules of inference of a first-order Peano Arithmetic, .

The misconception arises because actually admits *two* logical entailments, only one of which—Gödelian provability—has, so far, been formally acknowledged.

However, the other—familiar only in its avatar as the *intuitive* truth of a proposition under ‘s standard interpretation—*does*, also, follow *formally* from the axioms and rules of inference of .

Even when this issue is sought to be addressed, the argument is indirect, and this point remains implicit.

For instance, in a critical review of Roger Penrose’s Gödelian argument, Martin Davis (1990. *Is Mathematical Insight Algorithmic?* Behavioural and Brain Sciences, vol. 13 (1990), pp. 659–660) argues that:

“… There is an algorithm which, given any consistent set of axioms, will output a polynomial equation which in fact has no integer solutions, but such that this fact can not be deduced from the given axioms. Here then is the true but unprovable Gödel sentence on which Penrose relies and in a particularly simple form at that. Note that the sentence is provided by an algorithm. If insight is involved, it must be in convincing oneself that the given axioms are indeed consistent, since otherwise we will have no reason to believe that the Gödel sentence is true”.

Note that the first part of Gödel’s argument in Theorem VI of his 1931 paper is that, if is consistent, then we can *mechanically* construct a formula—which, syntactically, is of the form —such that:

(i) The formula , when viewed as a string of ‘meaningless’ symbols, *does not follow mechanically* from the axioms of as the last of any finite sequence of -formulas, each of which is either a -axiom, or a consequence of one or more of the formulas preceding it in the sequence, by the *mechanical* application of the rules of inference of ;

(ii) For any given numeral —which ‘represents’ the natural number in —the formula , when viewed as a string of ‘meaningless’ symbols, *does follow mechanically* from the axioms of as the last of some finite sequence of -formulas, each of which is either a -axiom, or a consequence of one or more of the formulas preceding it in the sequence, by the *mechanical* application of the rules of inference of .

Now, (i) is the standard definition (*due to Gödel*) of the meta-assertion:

(iii) The -formula is *formally* unprovable in .

However, under standard interpretations of Alfred Tarski’s definitions of the satisfiability and truth of the formulas of a language under an interpretation , the -formula is true in the interpretation if, and only if, the interpreted relation is instantiationally satisfied in (*i. e. for any given element of the interpreted relation can be ‘seen’ to hold in the interpretation*).

If we take both and as (as detailed in ‘*Evidence-Based Interpretations of PA*‘), and take satisfiability in to mean instantiational provability in , we arrive at the formal definition of the truth of the -formula in as:

The -formula is formally true in if, and only if, the formula is provable in whenever we substitute a numeral for the variable in .

Hence (ii) is the standard definition (*due to Tarski*) of the meta-assertion:

(iv) The -formula is *formally* true in .

So, by definition, the appropriate interpretation of Gödel’s reasoning (i) and (ii) ought to be:

(v) The -formula is *formally* unprovable in , but *formally* true in .

This interpretation also meets Ludwig Wittgenstein’s (*Remarks on the Foundations of Mathematics.* 1978 edition. MIT Press) requirement that the concept of ‘truth’ in a language must be formally definable, and effectively verifiable, within the language.

As noted by Reuben L. Goodstein (1972. *Wittgenstein’s Philosophy of Mathematics.* In Ambrose, Alice, and Morris Lazerowitz (eds.), *Ludwig Wittgenstein: Philosophy and Language.* George Allen and Unwin. pp. 271–86):

“In the realist-formalist controversy in the philosophy of mathematics Wittgenstein’s Remarks offers a solution that is crystal clear and satisfyingly uncompromising. The true propositions of mathematics are true because they are provable in a calculus; they are deductions from axioms by formal rules and are true in virtue of valid applications of the rules of inference and owe nothing to the world outside mathematics.”

However, standard expositions of Gödel’s formal reasoning assert only that:

(vi) The -formula is *formally* unprovable in , but *intuitively* true in the standard interpretation of .

They fail to highlight that, actually, (i) and (ii) are *both* logically entailed by the axioms and rules of inference of , and that, classically, the meta-assertion:

(vii) The -formula is *intuitively* true in the standard interpretation of .

is both ambiguous *and* stronger than the meta-assertion:

(viii) The -formula is *formally* true in .

The ambiguity surfaces in the presence of the Church-Turing Thesis, for (vii), then, implicitly implies that the arithmetical relation is algorithmically decidable as always true in the standard interpretation of , whereas (viii) does not.

**Use of square brackets**

Unless otherwise obvious from the context, we use square brackets to indicate that the contents represent a symbol or a formula—of a formal theory—generally assumed to be well-formed unless otherwise indicated by the context.

In other words, expressions inside the square brackets are to be only viewed syntactically as juxtaposition of symbols that are to be formed and manipulated upon strictly in accordance with specific rules for such formation and manipulation—in the manner of a mechanical or electronic device—without any regards to what the symbolism might represent semantically under an interpretation that gives them meaning.

**Use of an asterisk**

Unless otherwise obvious from the context, we use an asterisk to indicate that the associated expression is to be interpreted semantically with respect to some well-defined interpretation.

**Explanatory comments**

We have taken some liberty in emphasising standard definitions selectively, and interspersing our arguments liberally with comments and references, generally of a foundational nature.

These are intended to reflect our underlying thesis that essentially arithmetical problems appear more natural when expressed—and viewed—within an arithmetical perspective of an interpretation of PA that appeals to the evidence provided by a deterministic algorithm.

Since a deterministic algorithm has only one possible move from a given configuration such a perspective, by its very nature, cannot appeal implicitly to transfinite concepts.

**Evidence**

“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 …”

… 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.

**Aristotle’s particularisation**

This holds that from an assertion such as:

‘It is not the case that: For any given does not hold’

usually denoted symbolically by ‘‘, we may always validly infer in the classical, Aristotlean, logic of predicates ^{[1]} that:

‘There exists an unspecified such that holds’

usually denoted symbolically by ‘‘.

Aristotle’s particularisation (AP) is essentially the semantic postulation that from the negation of a universal we may always deduce the existence of a contrafactual. It is necessarily true over finite domains.

Expressed more formally:

**Aristotle’s particularisation under an interpretation**

If the formula of a first order language interprets as true under a sound interpretation of , then we may always conclude that there must be some object in the domain of the interpretation such that, if the formula interprets as the unary relation in , then the proposition is true under the interpretation.

**The significance of Aristotle’s particularisation for the first-order predicate calculus**

We note that in a formal language the formula ‘‘ is an abbreviation for the formula ‘‘.

The commonly accepted interpretation of this formula—and a fundamental tenet of classical logic unrestrictedly adopted as intuitively obvious by standard literature ^{[2]} 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 ^{[3]} 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 ‘‘ of a formal Arithmetical language interprets as an arithmetical relation denoted by ‘‘, and the formula ‘‘ as the arithmetical proposition denoted by ‘‘, the formula ‘‘ need not interpret as the arithmetical proposition denoted by the usual abbreviation ‘‘; and that such postulation is invalid as a general logical principle in the absence of a means for constructing some putative object for which the proposition holds in the domain of the interpretation.

Hence we shall follow the convention that the assumption that ‘‘ is the intended interpretation of the formula ‘‘—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 -consistency ^{[4]} as an explicit assumption in his formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions ^{[5]}.

Gödel explained at some length ^{[6]} that his reasons for introducing -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 -consistent if, and only if, Aristotle’s particularisation holds under the standard interpretation of PA ^{[7]}.

We note that Aristotle’s particularisation is a non-constructive—and logically fragile—semantic deduction rule. It is reflected in classical first order deduction either by some similarly non-constructive syntactic rule of natural deduction—such as Rosser’s *Rule C* ^{[7.1]}—or by the assumption that FOL is -consistent.

**The structure **

The structure of the natural numbers—namely:

(*the set of natural numbers*);

(*equality*);

(*the successor function*);

(*the addition function*);

(*the product function*);

(*the null element*).

**The axioms of the first-order Peano Arithmetic PA**

: ;

: ;

: ;

: ;

: ;

: ;

: ;

: ;

: For any well-formed formula of PA:

.

**Generalisation in PA**

If is PA-provable, then so is .

**Modus Ponens in PA**

If and are PA-provable, then so is .

**The standard interpretation of PA**

The standard interpretation of PA over the structure is the one in which the logical constants have their ‘usual’ interpretations ^{[8]} in Aristotle’s logic of predicates (which subsumes Aristotle’s particularisation), and ^{[9]}:

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

(b) the symbol interprets as the integer ;

(c) the symbol interprets as the successor operation (addition of );

(d) the symbols and interpret as ordinary addition and multiplication;

(e) the symbol interprets as the identity relation.

**Simple consistency**

A formal system S is simply consistent if, and only if, there is no S-formula for which both and are S-provable.

**-consistency**

A formal system S is -consistent if, and only if, there is no S-formula for which first is S-provable, and second is S-provable for any given S-term .

**Soundness (formal system – non-standard)**

A formal system S is sound under an interpretation with respect to a domain if, and only if, every theorem of S translates as ‘ is true under in ‘.

**Soundness (interpretation – non-standard)**

An interpretation of a formal system S is sound with respect to a domain if, and only if, S is sound under the interpretation over the domain .

**Soundness in classical logic**

In classical logic, a formal system is sometimes defined as ‘sound’ if, and only if, it has an interpretation; and an interpretation is defined as the assignment of meanings to the symbols, and truth-values to the sentences, of the formal system. Moreover, any such interpretation is defined as a model ^{[10]} of the formal system.

This definition suffers, however, from an implicit circularity: the formal logic underlying any interpretation of is implicitly assumed to be ‘sound’.

The above definitions seek to avoid this implicit circularity by delinking the defined ‘soundness’ of a formal system under an interpretation from the implicit ‘soundness’ of the formal logic underlying the interpretation.

This admits the case where, even if and are implicitly assumed to be sound, is sound, but is not.

Moreover, an interpretation of is now a model for if, and only if, it is sound.

**Algorithmic verifiability**

A number-theoretical relation is algorithmically verifiable if, and only if, for any given natural number , there is an algorithm which can provide objective evidence for deciding the truth/falsity of each proposition in the finite sequence .

**Tarskian interpretation of an arithmetical language verifiably in terms of the computations of a simple functional language**

We show in the Birmingham paper 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 if, and only if, they are algorithmically verifiable under the interpretation. ^{[11]}

**Algorithmic computability**

A number theoretical relation is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence for deciding the truth/falsity of each proposition in the denumerable sequence .

**Tarskian interpretation of an arithmetical language algorithmically in terms of the computations of a simple functional language**

We show in the Birmingham paper 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 if, and only if, they are algorithmically computable under the interpretation. ^{[12]}

**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 ^{[13]}, 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 ^{[14]}—and not only to a mathematically definable limit—if and only if such representation is definable by an algorithmically computable function. ^{[15]}

We note that although every algorithmically computable relation is algorithmically verifiable, the converse is not true. ^{[16]}

**References**

**Be59** Evert W. Beth. 1959. *The Foundations of Mathematics.* 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.

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

**BF58** Paul Bernays and Abraham A. Fraenkel. 1958. *Axiomatic Set Theory* 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.

**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.

**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.

**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.

**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.

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

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

**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 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.

**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 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: HA28, pp.58-59.

Return to 2: 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.

Return to 3: Br08.

Return to 4: The significance of -consistency for the formal system PA is highlighted in An12.

Return to 5: Go31, p.23 and p.28.

Return to 6: In his introduction on p.9 of Go31.

Return to 7: For details see An12.

Return to 7.1: See Ro53, pp.127-136.

Return to 8: See Me64, p.49.

Return to 9: See Me64, p.107.

Return to 10: We follow the definition in Me64, p.51.

Return to 11: We show in An12 that the concept of Algorithmic verifiability is also well-defined under the standard interpretation of PA over .

Return to 12: We show in An12 that the concepts of Algorithmic verifiability and Algorithmic computability are both well-defined under the standard interpretation of PA over ; moreover they identify distinctly different subsets of the well-defined PA formulas.

Return to 13: We note that the concept of ‘algorithmic computability’ is essentially an expression of the more rigorously defined concept of ‘realizability’ in Kl52, p.503.

Return to 14: In the sense of a physically ‘completable’ infinite sequence (as needed to resolve Zeno’s paradox).

Return to 15: This point is addressed in more detail in An13.

Return to 16: See Appendix B of this preprint *Is Gödel’s undecidable proposition an ‘ad hoc’ anomaly?*.

**Two perspectives on Hilbert’s First and Second Problems**

In the Birmingham paper ‘Evidence-Based Interpretations of PA’, we introduced the distinction between algorithmic verifiability and algorithmic computability.

We showed how the distinction naturally helped distinguish between a finitary algorithmic interpretation of PA, and the non-finitary standard interpretation of PA.

We then showed how the former yielded a finitary proof of consistency for PA, as demanded by the second of Hilbert’s celebrated 23 problems.

In the previous post, we also highlighted why the the non-finitary standard interpretation of PA does not yield a finitary proof of consistency for PA.

We now show that the difference between the conclusions suggested by finitary reasoning and those suggested by non-finitary reasoning in the previous pages (as in the case of Goodstein’s argument) is reflected further in the differing status of the Continuum Hypothesis (the first of Hilbert’s 23 problems) when viewed from finitary and non-finitary perspectives as detailed below.

**The non-finitary set-theoretical perspective**

The non-finitary set-theoretical perspective on the Continuum Hypothesis is well-known, and described succintly by Topologist Peter Nyikos in a short expository lecture given at the University of Auckland in May, 2000:

“In 1900, David Hilbert gave a seminal lecture in which he spoke about a list of unsolved problems in mathematics that he deemed to be of outstanding importance. The first of these was Cantor’s continuum problem, which has to do with infinite numbers with which Cantor revolutionised set theory. The smallest infinite number, , `aleph-nought,’ gives the number of positive whole numbers. A set is of this cardinality if it is possible to list its members in an arrangement such that each one is encountered after a finite number (however large) of steps. Cantor’s revolutionary discovery was that the points on a line cannot be so listed, and so the number of points on a line is a strictly higher infinite number (, `the cardinality of the continuum’) than . Hilbert’s First Problem asks whether any infinite subset of the real line is of one of these two cardinalities. The axiom that this is indeed the case is known as the Continuum Hypothesis (CH). …

Gödel [1940] also gave a partial solution to Hilbert’s First Problem by showing that the Continuum Hypothesis (CH) is consistent if the usual Zermelo-Fraenkel (ZF) axioms for set theory are consistent. He produced a model, known as the Constructible Universe, of the ZF axioms in which both the Axiom of Choice (AC) and the CH hold. Then Cohen showed in 1963 that the negations of these axioms are also consistent with ZF; in particular, CH can fail while AC holds in a model of ZF.”

… *Hilbert’s First and Second Problems and the foundations of mathematics*, Topology Atlas Document # taic-52, Topology Atlas Invited Contributions vol. 9, no. 3 (2004) 6 pp.

**Is CH a Deﬁnite Mathematical Problem?**

Well, the non-finitary set-theoretical formulation of the Continuum Hypothesis isn’t, according to Solomon Feferman who, in a presentation at the inaugural Paul Bernays Lectures, ETH, Zurich, Sept. 12, 2012, restated in his presentation that:

My view: No; in fact it is essentially indeﬁnite (“inherently vague”).

That is, the concepts of arbitrary set and function as used in its formulation even at the level of P(N) are essentially indeﬁnite.

… *Why isn’t the Continuum Problem on the Millennium ($1,000,000) Prize List?* CSLI Workshop on *Logic, Rationality and Intelligent Interaction*, Stanford, June 1, 2013.

Feferman sought to place in perspective the anti-Platonistic basis for his belief by quoting:

“Those who argue that the concept of set is not sufﬁciently clear to ﬁx the truth-value of CH have a position which is at present difﬁcult to assail. As long as no new axiom is found which decides CH, their case will continue to grow stronger, and our assertion that the meaning of CH is clear will sound more and more empty.”

… D. A. Martin, *Hilbert’s ﬁrst problem: The Continuum Hypothesis,* in *Mathematical Developments arising from Hilbert Problems,* Felix E. Browder, Rutgers University, Editor – American Mathematical Society, 1976, 628 pp.

**A finitary arithmetical perspective**

However a possible candidate for a finitary arithmetical perspective (as proposed in the previous pages of these investigations) is reflected in the following:

**Theorem:** There is no set whose cardinality is strictly between the cardinality of the integers and the cardinality of the real numbers.

**Proof:** By means of Gödel’s -function , we can show that if denotes the digit in the decimal expansion of a putatively given real number R in the interval then, for any given natural number , we can define an arithmetical function such that:

for all .

Since Gödel’s -function is primitive recursive, it follows that every putatively given real number R can be uniquely corresponded to an algorithmically verifiable arithmetical function within the first order Peano Arithmetic PA, where we define by:

for all ,

and is selected such that:

for all .

(*For the purist, the above conclusion can be justified by the argument in this preprint.*)

**Definition: Algorithmically verifiable function**

A number-theoretical function is algorithmically verifiable if, and only if, for any given natural number , there is an algorithm which can provide objective evidence for deciding the value of each formula in the finite sequence .

“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 …”.

… 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.

**Definition: Algorithmically computable function**

A number theoretical function is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence for deciding the value of each formula in the denumerable sequence .

**Cantor’s diagonal argument**

From a finitary arithmetical perspective, Cantor’s diagonal argument simply shows that there are algorithmically verifiable functions which are not algorithmically computable.

The correspondence is unique because, if R and S are two different putatively given reals in the interval , then there is always some for which . Hence we can always find corresponding arithmetical functions and such that:

for all .

for all .

.

Since PA is first order, the cardinality of the reals in the interval cannot, therefore, exceed that of the integers. The theorem follows.

In other words, the Continuum Hypothesis is trivially true from a finitary perspective because of the seemingly heretical conclusion that: , an answer that Hilbert would probably never have envisaged for the first of the celebrated twenty three problems that he bequethed to posterity!

**Skolem’s (apparent) paradox**

It is an answer that should, however, give comfort to the shades of Thoralf Skolem. In his 1922 address delivered in Helsinki before the Fifth Congress of Scandinavian Mathematicians, Skolem improved upon both the argument and statement of Löwenheim’s 1915 theorem—subsequently labelled as the:

**(Downwards) Löwenheim-Skolem Theorem**

If a first-order proposition is satisfied in any domain at all, then it is already satisfied in a denumerably infinite domain.

Skolem then cautioned about unrestrictedly (and meta-mathematically) corresponding putative mathematical entities across domains of different axiom systems, and drew attention to a:

“… peculiar and apparently paradoxical state of affairs. By virtue of the axioms we can prove the existence of higher cardinalities, of higher number classes, and so forth. How can it be, then, that the entire domain can already be enumerated by means of the finite positive integers? The explanation is not difficult to find. In the axiomatization, ‘set’ does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradiction at all if a set of the domain is non-denumerable in the sense of the axiomatization; for this means merely that *within* there occurs no one-to-one mapping of onto (Zermelo’s number sequence). Nevertheless there exists the possibility of numbering all objects in , and therefore also the elements of , by means of the positive integers; of course such an enumeration too is a collection of certain pairs, but this collection is not a ‘set’ (that is, it does not occur in the domain ).”

… Thoralf Skolem. 1922. *Some remarks on axiomatized set theory.* Text of an address delivered in Helsinki before the Fifth Congress of Scandinavian Mathematicians, 4-7 August 1922. In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts, p.295.

**What do you think?** Does the above argument apply to the finite ordinals? If so, is ZF inconsistent, or is it -consistent?

**Finitarily consistent mechanist reasoning and non-finitarily consistent human reasoning: Mutually inconsistent yet complementary!**

We now consider the following (tentatively expressed) conclusions suggested by our previous post, which we shall aim to investigate from various perspectives in these pages.

**Structures**

The Birmingham paper suggests that we may need to distinguish much more sharply than we do at present between:

Mathematical structures that are built upon only finitary reasoning, and

Mathematical structures that admit non-finitary reasoning.

**Interpretations**

For instance the Birmingham paper provides:

An example of a mathematical structure based on finitary reasoning, namely the finitarily sound algorithmic interpretation of the first order Peano Arithmetic PA.

An example of a mathematical structure based on non-finitary reasoning, namely the non-finitarily sound standard interpretation of the first order Peano Arithmetic PA.

The Birmingham paper suggests that the roots of the distinction between these two structures lies in the fact that:

Finitary reasoning does not assume that Aristotle’s particularisation is always true over infinite domains.

Non-finitary reasoning assumes that Aristotle’s particularisation is always true over infinite domains.

**Consistency of Arithmetic**

In the Birmingham paper we also show that:

Finitary reasoning proves that PA is consistent finitarily (as demanded by the second of Hilbert’s celebrated twenty three problems).

Non-finitary reasoning proves that PA is consistent non-finitarily (a consequence of Gentzen’s non-finitary proof of consistency for PA).

**FOL is consistent; FOL+AP is -consistent**

This suggests that:

Finitary reasoning as formalised in first order logic (FOL) is consistent.

Non-finitary reasoning as formalised in Hilbert’s -calculus (FOL+AP) is -consistent.

**-consistency**

Since the Birmingham paper shows that Aristotle’s particularisation holds over the structure of the natural numbers if, and only if, PA is -consistent, it suggests that:

Finitary reasoning does not admit that PA can be -consistent (see Corollary 4 of this post).

Non-finitary reasoning admits that PA can be -consistent.

**Arithmetical undecidability**

Since proofs of arithmetical undecidability implicitly assume Aristotle’s particularisation, this further suggests that:

Finitary reasoning does not admit undecidable arithmetical propositions (see Corollary 3 of this post).

Non-finitary reasoning admits undecidable arithmetical propositions.

**Completed Infinity**

A significant consequence is that:

Finitary reasoning does not admit an axiom of infinity.

Non-finitary reasoning admits an axiom of infinity.

**Non-standard models of PA**

A further consequence of this is that:

Finitary reasoning does not admit non-standard models of PA.

Non-finitary reasoning too does not admit non-standard models of PA.

**Algorithmically computable truth and algorithmically verifiable truth**

The Birmingham paper also suggests that:

The truths of finitary reasoning are algorithmically computable.

The truths of non-finitary reasoning are algorithmically verifiable, but not necessarily algorithmically computable.

**Categoricity and incompleteness of Arithmetic**

We show in Corollary 1 of this post that it also follows from the Birmingham paper that:

Finitary reasoning proves that PA is categorical with respect to algorithmically computable truth.

Non-finitary reasoning proves that PA is incomplete with respect to algorithmically verifiable truth (a consequence of Gödel’s proof of of the undecidability of some arithmetical propositions in any -consistent system of arithmetic).

**How intelligences reason**

This suggests that:

Finitary reasoning is a shared characteristic of all intelligences, human or non-human.

Non-finitary reasoning is a characteristic of human intelligence that may not be shared by any other intelligence.

**Communication between intelligences: SETI**

It further suggests that the search for extra-terrestrial intelligence may benefit from the argument that:

Finitary reasoning admits effective and unambiguous communication between two intelligences with respect to its (algorithmically computable) arithmetical truths.

Non-finitary reasoning does not admit effective and unambiguous communication between two intelligences with respect to its (algorithmically verifiable) arithmetical truths.

**Determinism, Unpredictability and the EPR paradox**

An unexpected consequence of the arguments of the Birmingham paper is that our perspectives on the relation between determinism and predictability may benefit from the paradigm shift demanded by the argument that:

Finitary reasoning admits the EPR paradox.

Non-finitary reasoning does not admit the EPR paradox.

The arguments of the Birmingham paper also suggest a fresh perspective on the issue of computationalism since:

Finitary reasoning does not admit Lucas’ Gödelian argument.

Non-finitary reasoning admits Lucas’ Gödelian argument.

**Effective computability**

It further suggests that the nature and status of ‘effective computability’ may also need to be assessed afresh since:

Finitary reasoning naturally equates algorithmic computability with effective computability.

Non-finitary reasoning naturally equates algorithmic verifiability with effective computability.

**Church Turing Thesis**

As also the nature of CT, since:

Finitary reasoning admits the Church-Turing Thesis.

Non-finitary reasoning does not admit the Church-Turing Thesis.

Broadly speaking, the two conflicting-but-complementary structures defined in the Birmingham paper suggest that we should be more explicit—in our argumentation—of the structure to which a particular assertion about the natural numbers pertains, since:

Both finitary and non-finitary reasoning do not admit the proof of Goodstein’s Theorem as neither admits a completed infinity.

Set-theoretical reasoning admits the proof of Goodstein’s Theorem as it admits a completed infinity.

**There’s more …**

In the next post we shall consider some further intriguing consequences suggested by the Birmingham paper.

**What do you think?**

Does Goodstein’s sequence over the natural numbers always terminate or not?

**Aristotle’s particularisation: A grey area in our accepted foundational concepts**

We shall now argue that what mathematics needs is not a new foundation, but a greater awareness of the nature of its existing foundations.

In particular, it is the thesis of these investigations that almost all of the unresolved philosophical issues in the foundations of mathematics reflect the fact that the nature and role of Aristotle’s particularisation is left implicit when it is postulated over infinite domains.

Perhaps that is the unintended consequence of ignoring Hilbert’s efforts to integrate the concept formally into first order logic by formally defining universal and existential quantification through the introduction of his -operator.

**Semantic postulation of Aristotle’s particularisation **

Aristotle’s particularisation (AP) is the postulation that from the negation of a universal we may always deduce the existence of a contrafactual.

(*It is necessarily true over finite domains.*)

More formally:

**Aristotle’s particularisation under an interpretation**

If the formula of a first order language interprets as true under a sound interpretation of , then we may always conclude that there must be some object in the domain of the interpretation such that, if the formula interprets as the unary relation in , then the proposition is true under the interpretation.

(*We note that Aristotle’s particularisation is a non-constructive—and logically fragile—semantic deduction rule. It is reflected in classical first order deduction either by some similarly non-constructive syntactic rule of natural deduction—such as Rosser’s Rule C—or by the assumption that FOL is -consistent.*)

**Is the price of Aristotle’s particularisation too high?**

If so, we shall argue that the price being asked for assuming AP implicitly—instead of explicitly as Hilbert had proposed—may be too high!

Partially because the assumption seems to effectively obscure the far-reaching consequences of the non-finitary nature of AP from immediate view in natural and formal deductive chains.

(*And therefore of the first order logic FOL under the implicit assumption of Aristotle’s particularisation.*)

For instance, as Carnap’s deduction of the Axiom of Choice in ZF illustrates, the non-finitary consequences of assuming AP over infinite domains becomes apparent when the underlying logic is taken as Hilbert’s -calculus instead of the classical first order logic FOL.

However, formal deductions apparently prefer to substitute—seemingly arbitrarily—the implicit assumption of AP in the underlying logic by the introduction of `contrived’ formal assumptions such as Gödel’s -consistency or Rosser’s Rule C.

**-consistency**

A formal system S is -consistent if, and only if, there is no S-formula for which, first, is S-provable and, second, is S-provable for any given S-term .

**Rosser’s Rule C**

“Since the rule ‘If , then ‘ corresponds to a hypothetical act of choice, we shall call it the rule of choice, or more briefly, Rule C.”

… J. Barkley Rosser. *Logic for Mathematicians.* 1953. McGraw Hill Book Company Inc., New York.

Similarly natural deduction chains apparently prefer to substitute—again seemingly arbitrarily—the implicit assumption of AP in the underlying logic by admitting a Rule of Infinite Induction (transfinite induction).

More importantly, the price may be too high because the implicit assumption of Aristotle’s particularisation in the underlying logic has masked the fact that, without such assumption, FOL is finitarily consistent; and—as we note below—that mathematically significant finitary structures can be built upon it without assuming AP.

**Evidence-Based Interpretations of PA**

Some consequences of making the assumption of Aristotle’s particularisation explicit are highlighted in `Evidence-Based Interpretations of PA’ that was presented at the AISB/IACAP Turing 2012 conference in Birmingham last year.

We showed there that Tarski’s inductive definitions admit evidence-based interpretations of the first-order Peano Arithmetic PA which allow us to define the satisfaction and truth of the quantified formulas of PA *constructively* over the domain of the natural numbers in *two* essentially different ways:

In terms of algorithmic verifiabilty; and

In terms of algorithmic computability.

That there can be even *one*, let alone *two*, logically sound (one finitary and one non-finitary) assignments of satisfaction and truth certificates to both the atomic and compound formulas of PA had hitherto been unsuspected!

**Definition: Algorithmically verifiable arithmetical truth**

A number-theoretical relation is algorithmically verifiable if, and only if, for any given natural number , there is an algorithm which can provide objective evidence for deciding the truth/falsity of each proposition in the finite sequence .

“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 …”.

… 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.

We show in the Birmingham paper (as we shall refer to it hereafter) 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 if, and only if, they are algorithmically verifiable under the interpretation.

**Definition: Algorithmically computable arithmetical truth**

A number theoretical relation is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence for deciding the truth/falsity of each proposition in the denumerable sequence .

We show in the Birmingham paper 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 if, and only if, they are algorithmically computable under the interpretation.

**Algorithmic verifiability vis à vis algorithmic computability**

We show in the Birmingham paper that the concepts of Algorithmic verifiability and Algorithmic computability are both well-defined under the standard interpretation of PA over ; moreover they identify distinctly different subsets of the well-defined PA formulas.

We show in this paper that although every algorithmically computable relation is algorithmically verifiable, the converse is not true.

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 (as addressed in more detail in this paper) 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.

**The finitarily sound algorithmic interpretation of PA over **

We argued from the above distinction that the algorithmically computable PA-formulas *can* provide a finitarily sound algorithmic interpretation of PA over the domain of the natural numbers.

We showed, moreover, that this yields a finitary proof of consistency for PA—as demanded by the Second of Hilbert’s celebrated Twenty Three Problems.

**The non-finitarily sound standard interpretation of PA over **

On the other hand, the distinction also suggests that Gerhard Gentzen’s transfinite proof of consistency for PA corresponds to the argument that the algorithmically verifiable PA-formulas of PA provide a non-finitarily sound standard interpretation of PA over .

Moreover—as has been generally suspected (perhaps for the reason noted towards the end of this post)—the distinction also suggests why the standard interpretation cannot yield the finitary proof of consistency for PA as demanded by Hilbert.

**The distinction between finitary and non-finitary arithmetical reasoning introduced in the Birmingham paper has far reaching consequences**

In these pages we shall argue that the power of this simple distinction actually goes far beyond the immediate conclusions drawn in the Birmingham paper.

Reason: We can further constructively define an unambiguous distinction between finitary and non-finitary reasoning, at the level of first order logic itself, which shows that the two are both mutually inconsistent yet complementary!

As can be expected, such a distinction could have far-reaching consequences for the foundations of mathematics, logic and computabiity (which form the focus of the investigations in these pages).

We shall consider some of these in the next post.

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