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Why primality is polynomial time, but factorisation is not

Differentiating between the signature of a number and its value

A brief review: The significance of evidence-based reasoning

In a paper: `The truth assignments that differentiate human reasoning from mechanistic reasoning: The evidence-based argument for Lucas’ Gödelian thesis’, which appeared in the December 2016 issue of Cognitive Systems Research [An16], I briefly addressed the philosophical challenge that arises when an intelligence—whether human or mechanistic—accepts arithmetical propositions as true under an interpretation—either axiomatically or on the basis of subjective self-evidence—without any specified methodology for objectively evidencing such acceptance in the sense of Chetan Murthy and Martin Löb:

“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: [Mu91], \S 1 Introduction.

“Intuitively we require that for each event-describing sentence, \phi_{o^{\iota}}n_{\iota} say (i.e. the concrete object denoted by n_{\iota} exhibits the property expressed by \phi_{o^{\iota}} ), there shall be an algorithm (depending on I, i.e. M^{*} ) to decide the truth or falsity of that sentence.” … Martin H Löb: [Lob59], p.165.

Definition 1 (Evidence-based reasoning in Arithmetic): Evidence-based reasoning accepts arithmetical propositions as true under an interpretation if, and only if, there is some specified methodology for objectively evidencing such acceptance.

The significance of introducing evidence-based reasoning for assigning truth values to the formulas of a first-order Peano Arithmetic, such as PA, under a well-defined interpretation (see Section 3 in [An16]), is that it admits the distinction:

(1) algorithmically verifiable `truth’ (Definition 2}); and

(2) algorithmically computable `truth’ (Definition 3).

Definition 2 (Deterministic algorithm): A deterministic algorithm computes a mathematical function which has a unique value for any input in its domain, and the algorithm is a process that produces this particular value as output.

Note that a deterministic algorithm can be suitably defined as a `realizer‘ in the sense of the Brouwer-Heyting-Kolmogorov rules (see [Ba16], p.5).

For instance, under evidence-based reasoning the formula [(\forall x)F(x)] of the first-order Peano Arithmetic PA must always be interpreted weakly under the classical, standard, interpretation of PA (see [An16], Theorem 5.6) in terms of algorithmic verifiability (see [An16], Definition 1); where, if the PA-formula [F(x)] interprets as an arithmetical relation F^{*}(x) over N :

Definition 2 (Algorithmic verifiability): The number-theoretical relation F^{*}(x) is algorithmically verifiable if, and only if, for any natural number n , there is a deterministic algorithm AL_{(F,\ n)} which can provide evidence for deciding the truth/falsity of each proposition in the finite sequence \{F^{*}(1), F^{*}(2), \ldots, F^{*}(n)\} .

Whereas [(\forall x)F(x)] must always be interpreted strongly under the finitary interpretation of PA (see [An16], Theorem 6.7) in terms of algorithmic computability ([An16], Definition 2), where:

Definition 3 (Algorithmic computability): The number theoretical relation F^{*}(x) is algorithmically computable if, and only if, there is a deterministic algorithm AL_{F} that can provide evidence for deciding the truth/falsity of each proposition in the denumerable sequence \{F^{*}(1), F^{*}(2), \ldots\} .

The significance of the distinction between algorithmically computable reasoning based on algorithmically computable truth, and algorithmically verifiable reasoning based on algorithmically verifiable truth, is that it admits the following, hitherto unsuspected, consequences:

(i) PA has two well-defined interpretations over the domain N of the natural numbers (including 0 ):

(a) the weak non-finitary standard interpretation I_{PA(N, SV)} ([An16], Theorem 5.6),

and

(b) a strong finitary interpretation I_{PA(N, SC)} ([An16], Theorem 6.7);

(ii) PA is non-finitarily consistent under I_{PA(N, SV)} ([An16], Theorem 5.7);

(iii) PA is finitarily consistent under I_{PA(N, SC)} ([An16], Theorem 6.8).

The significance of evidence-based reasoning for Computational Complexity

In this investigation I now show the relevance of evidence-based reasoning, and of distinguishing between algorithmically verifiable and algorithmically computable number-theoretic functions (as defined above), for Computational Complexity is that it assures us a formal foundation for placing in perspective, and complementing, an uncomfortably counter-intuitive entailment in number theory—Theorem 2 below—which has been treated by conventional wisdom as sufficient for concluding that the prime divisors of an integer cannot be proven to be mutually independent.

However, I show there that such informally perceived barriers are, in this instance, illusory; and that admitting the above distinction illustrates:

(a) Why the prime divisors of an integer are mutually independent Theorem 2;

(b) Why determining whether the signature (Definition 3 below) of a given integer n —coded as the key in a modified Bazeries-cylinder (see Definition 7 of this paper) based combination lock—is that of a prime, or not, can be done in polynomial time O(log_{_{e}}n) (Corollary 4 of this paper); as compared to the time \ddot{O}(log_{_{e}}^{15/2}n) given by Agrawal et al in [AKS04], and improved to \ddot{O}(log_{_{e}}^{6}n) by Lenstra and Pomerance in [LP11], for determining whether the value of a given integer n is that of a prime or not.

(c) Why it can be cogently argued that determining a factor of a given integer cannot be polynomial time.

Definition 4 (Signature of a number): The signature of a given integer n is the sequence a_{_{n,i}} where n + a_{_{n,i}} \equiv 0\ mod\ (p_{_{i}}) for all primes p_{_{i}}\ such\ that\ 1\leq i \leq \pi(\sqrt{n}) .

Unique since, if p_{_{\pi(\sqrt{m})+1}}^{2} > m \geq p_{_{\pi(\sqrt{m})}}^{2} and p_{_{\pi(\sqrt{n})+1}}^{2} > n \geq p_{_{\pi(\sqrt{n})}}^{2} have the same signature, then |m - n| = c_{_{1}}.\prod_{i=1}^{\pi(\sqrt{m})}p_{_{i}} = c_{_{2}}.\prod_{i=1}^{\pi(\sqrt{n})}p_{_{i}} ; whence c_{_{1}} = c_{_{2}} = 0 since \prod_{i=1}^{k}p_{_{i}} > (\prod_{i=2}^{k-2}p_{_{i}}).p_{_{k}}^{^{2}} > p_{_{k+1}}^{2} for k > 4 by appeal to Bertrand’s Postulate 2.p_{_{k}} > p_{_{k+1}} ; and the uniqueness is easily verified for k \leq 4 .

Definition 5 (Value of a number): The value of a given integer n is any well-defined interpretation—over the domain of the natural numbers—of the (unique) numeral [n] that represents n in the first-order Peano Arithmetic PA.

We note that Theorem 2 establishes a lower limit for [AKS04] and [LP11], because determining the signature of a given integer n does not require knowledge of the value of the integer as defined by the Fundamental Theorem of Arithmetic.

Theorem 1: (Fundamental Theorem of Arithmetic): Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}=\prod _{i=1}^{k}p_{i}^{n_{i}}

where p_{1} < p_{2} < \ldots < p_{k} are primes and the n_{i} are positive integers (including 0 ).

Are the prime divisors of an integer mutually independent?

In this paper I address the query:

Query 1: Are the prime divisors of an integer n mutually independent?

Definition 6 (Independent events): Two events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of the other.

Intuitively, the prime divisors of an integer seem to be mutually independent by virtue of the Fundamental Theorem of Arithmetic

Moreover, the prime divisors of n can also be seen to be mutually independent in the usual, linearly displayed, Sieve of Eratosthenes, where whether an integer n is crossed out as a multiple of a prime p is obviously independent of whether it is also crossed out as a multiple of a prime q \neq p :

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 …

Despite such compelling evidence, conventional wisdom appears to accept as definitive the counter-intuitive conclusion that although we can see it as true, we cannot mathematically prove the following proposition as true:

Proposition 1: Whether or not a prime p divides an integer n is independent of whether or not a prime q \neq p divides the integer n .

We note that such an unprovable-but-intuitively-true conclusion makes a stronger assumption than that in Gödel’s similar claim for his arithmetical formula [(\forall x)R(x)] —whose Gödel-number is 17Gen\ r —in [Go31], p.26(2). Stronger, since Gödel does not assume his proposition to be intuitively true, but shows that though the arithmetical formula with Gödel-number 17Gen\ r is not provable in his Peano Arithmetic P yet, for any P -numeral [n] , the formula [R(n)] whose Gödel-number is Sb \left(r \begin{array}{c}17 \\ Z(n)\end{array}\right) is P -provable, and therefore meta-mathematically true under any well-defined Tarskian interpretation of P (cf., [An16], Section 3.).

Expressed in computational terms (see [An16], Corollary 8.3), under any well-defined interpretation of P , Gödel’s formula [R(x)] translates as an arithmetical relation, say R'(x) , such that R'(n) is algorithmically verifiable, but not algorithmically computable, as always true over N , since [\neg (\forall x)R(x)] is P -provable ([An16], Corollary 8.2).

We thus argue that a perspective which denies Proposition 1 is based on perceived barriers that reflect, and are peculiar to, only the argument that:

Theorem 2: There is no deterministic algorithm that, for any given n , and any given prime p \geq 2 , will evidence that the probability \mathbb{P}(p\ |\ n) that p divides n is \frac{1}{p} , and the probability \mathbb{P}(p\not|\ n) that p does not divide n is 1 - \frac{1}{p} .

Proof By a standard result in the Theory of Numbers ([Ste02], Chapter 2, p.9, Theorem 2.1, we cannot define a probability function for the probability that a random n is prime over the probability space (1, 2, 3, \ldots, ) .

(Compare with the informal argument in [HL23], pp.36-37.)

In other words, treating Theorem 2 as an absolute barrier does not admit the possibility—which has consequences for the resolution of outstanding problems in both the theory of numbers and computational complexity—that Proposition 1 is algorithmically verifiable, but not algorithmically computable, as true, since:

Theorem 3: For any given n , there is a deterministic algorithm that, given any prime p \geq 2 , will evidence that the probability \mathbb{P}(p\ |\ n) that p divides n is \frac{1}{p} , and the probability \mathbb{P}(p\not|\ n) that p does not divide n is 1 - \frac{1}{p} .

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

Can Gödel be held responsible for not clearly distinguishing—in his seminal 1931 paper on formally undecidable propositions (pp.596-616, ‘From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931‘, Jean van Heijenoort, Harvard University Press, 1976 printing)—between the implicit circularity that is masked by the non-constructive nature of his proof of undecidability in PM, and the lack of any circularity in his finitary proof of undecidability in Peano Arithmetic?

“The analogy of this argument with the Richard antinomy leaps to the eye. It is closely related to the “Liar” too;[Fn.14] for the undecidable proposition [R (q); q] states that q belongs to K , that is, by (1), that [R (q); q] is not provable. We therefore have before us a proposition that says about itself that it is not provable [in PM].[Fn.15]

[Fn.14] Any epistemological antinomycould be used for a similar proof of the existence of undecidable propositions.”

[Fn.15] Contrary to appearances, such a proposition involves no faulty circularity, for initially it [only] asserts that a certain well-defined formula (namely, the one obtained from the q th formula in the lexicographic order by a certain substitution) is unprovable. Only subsequently (and so to speak by chance) does it turn out that this formula is precisely the one by which the proposition itself was expressed.”

It is a question worth asking, if we heed Abel-Luis Peralta, who is a Graduate in Scientific Calculus and Computer Science in the Faculty of Exact Sciences at the National University of La Plata in Buenos Aires, Argentina; and who has been contending in a number of posts on his Academia web-page that:

(i) Gödel’s semantic definition of ‘[R(n) : n] ‘, and therefore of ‘\neg Bew[R(n) : n] ‘, is not only:

(a) self-referential under interpretation—in the sense of the above quote (pp.597-598, van Heijenoort) from Gödel’s Introduction in his 1931 paper ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems I’ (pp.596-616, van Heijenoort);

but that:

(b) neither of the definitions can be verified by a deterministic Turing machine as yielding a valid formula of PM.

Peralta is, of course, absolutely right in his contentions.

However, such non-constructiveness is a characteristic of any set-theoretical system in which PM is interpretable; and in which, by Gödel’s self-confessed Platonism (apparent in his footnote #15 in the quote above), we do not need to establish that his definitions of ‘[R(n) : n] ‘ and ‘\neg Bew[R(n) : n] ‘ need to be verifiable by a deterministic Turing machine in order to be treated as valid formulas of PM.

Reason: By the usual axiom of separation of any formal set theory such as ZFC in which PM is interpreted, Gödel’s set-theoretical definition (p.598, Heijenoort):

n \in K \equiv \neg Bew[R(n) : n]

lends legitimacy to \neg Bew[R(n) : n] as a PM formula.

Thus Gödel can formally assume—without further proof, by appeal simply to the axiom of choice of ZFC—that the PM formulas with exactly one variable—of the type of natural numbers—can be well-ordered in a sequence in some way such as, for example (Fn.11, p.598, Heijenoort):

“… by increasing the sum of the finite sequences of integers that is the ‘class sign’;, and lexicographically for equal sums.”

We cannot, though, conclude from this that:

(ii) Gödel’s formally undecidable P-formula, say [(\forall x)R(x)] —whose Gödel-number is defined as 17Gen\ r in Gödel’s proof of his Theorem VI (on pp.607-609 of van Heijenoort)—also cannot be verified by a deterministic Turing machine to be a valid formula of Gödel’s Peano Arithmetic P.

Reason: The axioms of set-theoretical systems such as PM, ZF, etc. would all admit—under a well-defined interpretation, if any—infinite elements, in the putative domain of any such interpretation, which are not Turing-definable.

Nevertheless, to be fair to two generations of scholars who—apart from those who are able to comfortably wear the logician’s hat—have laboured in attempts to place the philosophical underpinnings of Gödel’s reasoning (in his 1931 paper) in a coherent perspective (see this post; also this and this), I think Gödel must, to some extent, be held responsible—but in no way accountable—for the lack of a clear-cut distinction between the non-constructivity implicit in his semantic proof in (i), and the finitarity that he explicitly ensures for his syntactic proof in (ii).

Reason: Neither in his title, nor elsewhere in his paper, does Gödel categorically state that his goal was:

(iii) not only to demonstrate the existence of formally undecidable propositions in PM, a system which admits non-finitary elements under any putative interpretation;

(iv) but also to prevent the admittance of non-finitary elements—precisely those which would admit conclusions such as (ii)—when demonstrating the existence of formally undecidable propositions in ‘related’ systems such as his Peano Arithmetic P.

He merely hints at this by stating (see quote below from pp.587-589 of van Heijenoort) that his demonstration of (iii) is a ‘sketch’ that lacked the precision which he intended to achieve in (iv):

“Before going into details, we shall first sketch the main idea of the proof, of course without any claim to complete precision. The formulas of a formal system (we restrict ourselves here to the system PM) in outward appearance are finite sequences of primitive signs (variables, logical constants, and parentheses or punctuation dots), and it is easy to state with complete precision which sequences of primitive signs are meaningful formulas and which are not….

by:

(v) weakening the implicit assumption—of the decidability of the semantic truth of PM-propositions under any well-defined interpretation of PM—which underlies his proof of the existence of formally undecidable set-theoretical propositions in PM;

The method of proof just explained can clearly be applied to any formal system that, first, when interpreted as representing a system of notions and propositions, has at its disposal sufficient means of expression to define the notions occurring in the argument above (in particular, the notion “provable formula”) and in which, second, every provable formula is true in the interpretation considered. The purpose of carrying out the above proof with full precision in what follows is, among other things, to replace the second of the assumptions just mentioned by a purely formal and much weaker one.”

and:

(vi) insisting—in his proof of the existence of formally undecidable arithmetical propositions in his Peano Arithmetic P—upon the introduction of a methodology for constructively assigning unique truth values to only those (primitive recursive) quantified number-theoretic assertions (#1 to #45 on pp.603-606 of van Heijenoort) that are bounded when interpreted over the domain N of the natural numbers (footnote #34 on p.603 of van Heijenoort):

“Wherever one of the signs (x) , (Ex) , or \varepsilon x occurs in the definitions below, it is followed by a bound on x . This bound serves merely to ensure that the notion defined is recursive (see Theorem IV). But in most cases the extension of the notion defined would not change if this bound were omitted.”

From today’s perspective, one could reasonably hold that—as Peralta implicitly contends—Gödel is misleadingly suggesting (in the initial quote above from pp.587-589 of van Heijenoort) that his definitions of ‘[R(n) : n] ‘ and ‘~Bew[R(n) : n] ‘ may be treated as yielding ‘meaningful’ formulas of PM which are well-definable constructively (in the sense of being definable by a deterministic Turing machine).

In my previous post I detailed precisely why such an assumption would be fragile, by showing how the introduction of the boundedness Gödel insisted upon in (vi) distinguishes:

(vii) Gödel’s semantic proof of the existence of formally undecidable set-theoretical propositions in PM (pp.598-599 of van Heijenoort), which admits Peralta’s contention (1);

from:

(viii) Gödel’s syntactic proof of the existence of formally undecidable arithmetical propositions in the language of his Peano Arithmetic P (pp.607-609 of van Heijenoort), which does not admit the corresponding contention (ii).

Moreover, we note that:

(1) Whereas Gödel can—albeit non-constructively—claim that his definition of ‘Bew[R(n) : n] ‘ yields a formula in PM, we cannot claim, correspondingly, that his primitive recursive formula Bew(x) is a formula in his Peano Arithmetic P.

(2) The latter is a number-theoretic relation defined by Gödel in terms of his primitive recursive relation #45, ‘xBy ‘, as:

#46. Bew(x) \equiv (\exists y)yBx .

(3) In Gödel’s terminology, ‘Bew(x) ‘ translates under interpretation over the domain N of the natural numbers as:

x is the Gödel-number of some provable formula [F] of Gödel’s Peano Arithmetic P’.

(4) However, unlike Gödel’s primitive recursive functions and relations #1 to #45, both ‘(\exists y)yBx ‘ and ‘\neg (\exists y)yBx ‘ are number-theoretic relations which are not primitive recursive—which means that they are not effectively decidable by a Turing machine under interpretation in N.

(5) Reason: Unlike in Gödel’s definitions #1 to #45 (see footnote #34 on p.603 of van Heijenoort, quoted above), there is no bound on the quantifier ‘(\exists y) ‘ in the definition of Bew(x) .

Hence, by Turing’s Halting Theorem, we cannot claim—in the absence of specific proof to the contrary—that there must be some deterministic Turing machine which will determine whether or not, for any given natural number m , the assertion Bew(m) is true under interpretation in N.

This is the crucial difference between Gödel’s semantic proof of the existence of formally undecidable set-theoretical propositions in PM (which admits Peralta’s contention (i)), and Gödel’s syntactic proof of the existence of formally undecidable arithmetical propositions in the language of his Peano Arithmetic P (which does not admit his contention (i)).

(6) We cannot, therefore—in the absence of specific proof to the contrary—claim by Gödel’s Theorems V or VII that there must be some P-formula, say [Bew_{_{PA}}(x)] (corresponding to the PM-formula Bew[R(n) : n] ), such that, for any given natural number m :

(a) If Bew(m) is true under interpretation in N, then [Bew_{_{PA}}(m)] is provable in P;

(b) If \neg Bew(m) is true under interpretation in N, then \neg [Bew_{_{PA}}(m)] is provable in P.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

A: Is Gödel’s reasoning really kosher?

Many scholars yet harbour a lingering suspicion that Gödel’s definition of his formally undecidable arithmetical proposition [(\forall x)R(x,p)] involves a latent contradiction—arising from a putative, implicit, circular self-reference—that is masked by unverifiable, even if not patently invalid, mathematical reasoning.

The following proof of Gödel’s Theorem VI of his 1931 paper is intended to:

\bullet strip away the usual mathematical jargon that shrouds proofs of Gödel’s argument which makes his—admittedly arcane—reasoning difficult for a non-logician to unravel;

and

\bullet show that, and why—unlike in the case of the paradoxical ‘Liar’ sentence: ‘This sentence is a lie’—Gödel’s proposition [(\forall x)R(x, p)] does not involve any circular self-reference that could yield a Liar-like contradiction, either in a formal mathematical language, or when interpreted in any language of common discourse.

B: Gödel’s 45 primitive recursive arithmetic functions and relations

We begin by noting that:

(1) In his 1931 paper on formally ‘undecidable’ arithmetical propositions, Gödel shows that, given a well-defined system of Gödel-numbering, every formula of a first-order Peano Arithmetic such as PA can be Gödel-numbered by Gödel’s primitive recursive relation #23, Form(x) , which is true if, and only if, x is the Gödel-number (GN) of a formula of PA.

(2) So, given any natural number n , (1) allows us to decompose n and effectively determine whether, or not, n is the GN of some PA formula.

(3) Gödel also defines a primitive recursive relation #44, Bw(x) , which is true if, and only if, x is the GN of a finite sequence of formulas in PA, each of which is either an axiom, or an immediate consequence of two preceding formulas in the sequence.

(4) So, given any natural number n , (3) allows us to effectively determine whether, or not, the natural number n is the GN of a proof sequence in PA.

(5) Further, Gödel defines a primitive recursive relation #45, xBy , which is true if, and only if, x is the GN of a proof sequence in PA whose last formula has the GN y .

(6) Gödel then defines a primitive recursive relation, say Q(x,y) \equiv xBSUBy , such that, for any m,n :

mBSUBn is true if, and only if, m happens to be a GN that can be decomposed into a proof sequence whose last member is some PA formula [F(n)] , and n happens to be a GN that decomposes into the PA-formula [F(u)] with only one variable [u] .

(7) The essence of Gödel’s Theorem VI lies in answering the question:

Query 1: Is there any natural number n for which mBSUBn is true?

C: Gödel’s reasoning in Peano Arithmetic

(8) Now, by Gödel’s Theorem VII (a standard representation theorem of arithmetic), xBSUBy can be expressed in PA by some (formally well-defined) formula [\neg R(x,y)] such that, for any m,n :

(a) If mBSUBn is true, then [\neg R(m,n)] is PA-provable;

(b) If \neg mBSUBn is true, then [R(m,n)] is PA-provable.

(9) Further, by (6) and (8), for any m,n , if n is the GN of F(x) then:

(a) If mBSUBn is true, then [R(m,n)] is PA-provable; and m is a PA-proof of [F(n)] ;

(b) If \neg mBSUBn is true, then [\neg R(m,n)] is PA-provable; and m is not a PA-proof of [F(n)] .

(10) In his Theorem VI, Gödel then argues as follows:

(a) Let q be the GN of the formula [R(x,y)] defined in (8).

(b) Let p be the GN of [(\forall x)R(x,y)] .

(c) Let r be the GN of [R(x,p)] .

(d) Let 17Gen\ r be the GN of [(\forall x)R(x,p)] .

(11) We note that all the above primitive recursive relations are formally well-defined within the standard primitive recursive arithmetic PRA; and all the PA-formulas—as well as their corresponding Gödel-numbers—are well-defined in the first-order Peano Arithmetic PA.

In other words, as Gödel emphasised in his paper, the 46—i.e., 45 + xBSUBy —PRA functions and relations that he defines are all bounded, and therefore effectively decidable as true or false over the domain N of the natural numbers; whilst the PA-formulas that he defines do not involve any reference—or self-reference—to either the meaning or the truth/falsity of any PA-formulas under an interpretation in N , but only to their PA-provability which, he shows, is effectively decidable by his system of Gödel-numbering and his definition of the primitive recursive relation xBy .

(12) If we now substitute p for n , and [(\forall x)R(x,p)] for [F(n)] , in (9) we have (since p is the GN of [(\forall x)R(x,y)] ) that:

(i) If mBSUBp is true, then [R(\neg m,p)] is PA-provable; whence m is a PA-proof of [(\forall x)R(x,p)] ;

(ii) If \neg mBSUBp is true, then [R(m,p)] is PA-provable; whence m is not a PA-proof of [(\forall x)R(x,p)] .

Hence n = p answers Query 1 affirmatively.

D: Gödel’s conclusions

(13) Gödel then concludes that, if PA is consistent then:

By (12)(i), if mSUBp is true for some m , then both [R(\neg m,p)] and [(\forall x)R(x,p)] are PA-provable—a contradiction since, by Generalisation in PA, the latter implies that [R(m,p)] is provable in PA.

Hence [(\forall x)R(x,p)] , whose GN is 17Gen\ r , is not provable in PA if PA is consistent.

(14) Moreover, if PA is assumed to also be \omega -consistent (which means that we cannot have a PA-provable formula [\neg (\forall x)F(x)] such that [F(m)] is also provable in PA for any given numeral [m] ) then:

By (13), m is not a PA-proof of [(\forall x)R(x,p)] for any given m ; whence [R(m,p)] is PA-provable for any given m by (12)(ii).

Hence [\neg (\forall x)R(x,p)] , whose GN is Neg(17Gen r) , is not provable in PA.

E: Gödel’s [(\forall x)R(x,p)] does not refer to itself

We note that Gödel’s formula [(\forall x)R(x,p)] —whose GN is 17Gen\ r — does not refer to itself since it is defined in terms of the natural number p , and not in terms of the natural number 17Gen\ r .

F: Somewhere, far beyond Gödel

The consequences of Gödel’s path-breaking answer to Query 1 are far-reaching (as detailed in this thesis).

For instance, taken together with the proof that PA is categorical with respect to algorithmic computability (Corollary 7.2 of this paper), and that PA is not \omega -consistent (Corollary 8.4 of this paper), the above entails that:

\bullet There can be no interpretation of Gödel’s definition of his formally undecidable arithmetical proposition [(\forall x)R(x),p] over the domain of the natural numbers—whether expressed mathematically or in any language of common discourse—that could lead to a contradiction;

\bullet Gödel’s [(\forall x)R(x,p)] is not a formally undecidable arithmetical proposition, since [\neg (\forall x)R(x,p)] is PA-provable (see Corollary 8.2 of this paper).

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

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

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

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

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

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

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

The Brains blog

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

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

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

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

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

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

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

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

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

III: The Unexplained Intellect: The importance of computability

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

IV: The Unexplained Intellect: Consequences of imperfection

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

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

V: The Unexplained Intellect: The importance of rapport

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

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

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

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

A: Simplifying Mole’s perspective

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

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

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

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

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

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

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

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

B. Support for Mole’s thesis

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

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

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

The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas’ Goedelian Thesis

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

C. Algorithmic computability

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

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

D. Algorithmic verifiability

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Algorithmically Verifiable Logic vis `a vis Algorithmically Computable Logic: Could resolving EPR need two complementary Logics?

(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 \S4 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):

Why Hilbert’s and Brouwer’s interpretations of quantification are complementary and not contradictory.’

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

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

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

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

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

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

The question arises:

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

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

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

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

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

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

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

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

(i) is finitarily consistent; but

(ii) is not \omega-consistent; and

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

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

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

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

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

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

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

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

A distinction that is reflected in Stephen G. Simpson’s more conservative perspective in Partial Realizations of Hilbert’s Program (\S6.4, p.15):

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

Reason:

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

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

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

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

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

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

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

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

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

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

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

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

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

\S 1 The Logical Issue

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

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

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

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

(i) The concept of algorithmic verifiability is well-defined under the standard interpretation \mathcal{I}_{PA(\mathbb{N},\ Standard)} of PA over the structure \mathbb{N} of the natural numbers; and

(ii) The concept of algorithmic computability too is well-defined under the algorithmic interpretation \mathcal{I}_{PA(\mathbb{N},\ Algorithmic)} of PA over the structure \mathbb{N} 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.

\S 2 Weakening the Church and Turing Theses

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

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

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

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

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

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

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

Why Church’s Thesis?

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

Kurt Gödel’s reservations

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

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

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

Alan Turing’s perspective

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

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

Boolos, Burgess and Jeffrey’s query

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

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

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

However, as Boolos, Burgess and Jeffrey note quizically:

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

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

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

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

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

\S 2.2 Distinguishing between algorithmic verifiability and algorithmic computability

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

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

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

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

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

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

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

\S 2.2.2 Defining effective computability

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

(i) Algorithmically verifiable;

(ii) Algorithmically computable.

under an interpretation.

We shall thus propose the definition:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(ii) c = j!;

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

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

Now we have the standard definition [22]:

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

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

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

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

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

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

Formally, however, the PA formula:

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

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

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

We then have:

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

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

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

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

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

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

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

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

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

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

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

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

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

We thus have that:

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

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

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

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

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

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

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

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

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

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

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

and:

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

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

We now see that:

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

Proof: It follows from Lemma 4 that:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The theorem follows. \Box

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

The above theorem now suggests the following definition:

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

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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, \S9(II), p.139.

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

Return to 11: BBJ03, p. 37.

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

Return to 13: BBJ03, p.37.

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

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

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

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

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

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

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

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

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

Return to 22: Me64, p.118.

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

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

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

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

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

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

So where exactly does the buck stop?

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

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

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

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

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

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

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

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

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

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

Clearly, the debatable issue is the third case.

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

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

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

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

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

However, CT does not succeed in its objective completely.

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

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

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

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

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

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

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

See also (i) this later publication by Sebastian Grève, where he concludes that “… while Gödel indeed showed some significant understanding of Wittgenstein here, ultimately, Wittgenstein perhaps understood Gödel better than Gödel understood himself”; and (ii) this note on Rosser’s Rule C and Wittgenstein’s objections on purely philosophical considerations to Gödel’s reasoning and conclusions, where we show that, although not at all obvious (perhaps due to Gödel’s overpoweringly plausible presentation of his interpretation of his own formal reasoning over the years) what Gödel claimed to have proven is not—as suspected and held by Wittgenstein—supported by Gödel’s formal argumentation.

A: Misunderstanding Gödel’s Theorems VI and XI: Incompleteness and Undecidability

In an informal essay, “Misunderstanding Gödel’s Theorems VI and XI: Incompleteness and Undecidability“, DPhil candidate Sebastian Grève at The Queen’s College, Oxford, attempts to come to terms with what he subjectively considers:

“… has not been properly addressed as such by philosophers hitherto as of great philosophical importance in our understanding of Gödel’s Incompleteness Theorems.”

Grève’s is an unusual iconoclastic perspective:

“This essay is an open enquiry towards a better understanding of the philosophical significance of Gödel’s two most famous theorems. I proceed by a discussion of several common misunderstandings, led by the following four questions:

1) Is the Gödel sentence true?

2) Is the Gödel sentence undecidable?

3) Is the Gödel sentence a statement?

4) Is the Gödel sentence a sentence?

Asking these questions in this order means to trace back the steps of Gödel’s basic philosophical interpretation of his formal results. What I call the basic philosophical interpretation is usually just taken for granted by philosopher’s writing about Gödel’s theorems.”

In a footnote Grève acknowledges Wittgenstein’s influence by suggesting that:

“This essay can be read as something like a free-floating interpretation of the theme of Wittgenstein’s remarks on Gödel’s Incompleteness Theorems in Wittgenstein: 1978[RFM], I-(III), partly following Floyd: 1995 but especially Kienzler: 2008, and constituting a reply to inter alia Rodych: 2003”.

B: Why we may see the trees, but not the forest

We note that Grève’s four points are both overdue and well-made:

1. Is the Gödel sentence true?

Grève’s objection that standard interpretations are obscure when they hold the Gödel sentence as being intuitively true deserves consideration (see this post).

The ‘truth’ of the sentence should and does—as Wittgenstein stressed and suggested—follow objectively from the axioms and rules of inference of arithmetic.

2. Is the Gödel sentence undecidable?

Grève’s observation that the ‘undecidability’ of the Gödel sentence conceals a philosophically questionable assumption is well-founded.

The undecidability in question follows only on the assumption of ‘\omega-consistency’ made explicitly by Gödel.

This assumption is actually logically equivalent to the philosophically questionable assertion that from the provability of [\neg(\forall x)R(x)] we may conclude the existence of some numeral [n] for which [R(n)] is provable.

Since Rosser’s proof implicitly makes this assumption by means of his logically questionable Rule C, his claim of avoiding omega-consistency for arithmetic is illusory.

3. Is the Gödel sentence a statement?

Grève rightly holds that the Gödel sentence should be treated as a valid statement within the formal arithmetic S, since it is structurally defined as a well-formed formula of S.

4. Is the Gödel sentence a sentence?

Grève’s concern about whether the Gödel sentence of S is a valid arithmetical proposition under interpretation also seems to need serious philosophical consideration.

It can be argued (see the comment following the proof of Lemma 9 of this preprint) that the way the sentence is formally defined as the universal quantification of an instantiationally (but not algorithmically) defined arithmetical predicate does not yield an unequivocally defined arithmetical proposition in the usual sense under interpretation.

In this post [*] we shall not only echo Grève’s disquietitude, but argue further that Gödel’s interpretation and assessment of his own formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions is, essentially, a post-facto imposition that continues to influence standard expositions of Gödel’s reasoning misleadingly.

Feynman’s cover-up factor

Our thesis is influenced by physicist Richard P. Feynman, who started his 1965 Nobel Lecture with a penetrating observation:

We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn’t any place to publish, in a dignified manner, what you actually did in order to get to do the work.

That such `cover up’ may have unintended—and severely limiting—consequences on a discipline is suggested by Gödel’s interpretation, and assessment, of his own formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions (Go31).

Thus, in his informal preamble to the result that he intended to prove formally, Gödel wrote (cf. Go31, p.9):

The analogy of this result with Richard’s antinomy is immediately evident; there is also a close relationship with the Liar Paradox … Thus we have a proposition before us which asserts its own unprovability.

Further, interpreting the significance of his formal reasoning as having established the existence of a formally undecidable arithmetical proposition that is, however, decidable by meta-mathematical arguments, Gödel noted that:

The precise analysis of this remarkable circumstance leads to surprising results concerning consistency proofs of formal systems … (Go31, p.9)

The true reason for the incompleteness which attaches to all formal systems of mathematics lies, as will be shown in Part II of this paper, in the fact that the formation of higher and higher types can be continued into the transfinite (c.f., D. Hilbert, `Über das Unendliche’, Math. Ann. 95, p. 184), while, in every formal system, only countable many are available. Namely, one can show that the undecidable sentences which have been constructed here always become decidable through adjunction of suitable high types (e.g. of the type \omega to the system P. A similar result also holds for the axiom systems of set theory. (Go31, p.28, footnote 48a)

The explicit thesis of this foundational paper is that the above interpretation is an instance of a `cover up’—in Feynman’s sense—which appears to be a post-facto imposition that, first, continues to echo in and misleadingly [1] influence standard expositions of Gödel’s reasoning when applied to a first-order Peano Arithmetic, PA, and, second, that it obscures the larger significance of the genesis of Gödel’s reasoning.

As Gödel’s various remarks in Go31 suggest, this possibly lay in efforts made at the dawn of the twentieth century—largely as a result of Brouwer’s objections (Br08)—to define unambiguously the role that the universal and existential quantifiers played in formal mathematical reasoning.

That this issue is critical to Gödel’s reasoning in Go31, but remains unresolved in it, is obscured by his powerful presentation and interpretation.

So, to grasp the underlying mathematical significance of Gödel’s reasoning, and of what he has actually achieved, one may need to avoid focusing (as detailed in the previous posts on A foundational perspective on the semantic and logical paradoxes; in this post on undecidable Gödelian propositions, and in this preprint on undecidable Gödelian propositions):

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

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

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

\bullet on his assessment and interpretation of the formal consequences of such `incompleteness’.

We show in this paper that, when applied to PA [2], all of these obscure the deeper significance of what Gödel actually achieved in Go31.

C: Hilbert: If the \omega -Rule is true, can P be completed?

Instead, Gödel’s reasoning may need to be located specifically in the context of Hilbert’s Program (cf. Hi30, pp.485-494) in which he proposed an \omega-rule as a finitary means of extending a Peano Arithmetic—such as his formal system P in Go31—to a possible completion (i.e. to logically showing that, given any arithmetical proposition, either the proposition, or its negation, is formally provable from the axioms and rules of inference of the extended Arithmetic).

Hilbert’s \omega-Rule: If it is proved that the P-formula [F(x)] interprets as a true numerical formula for each given P-numeral [x], then the P-formula [(\forall x)F(x)] may be admitted as an initial formula (axiom) in P.

It is likely that Gödel’s 1931 paper evolved out of attempts to prove Hilbert’s \omega -rule in the limited—and more precise—sense that if a formula [F(n)] is provable in P for each given numeral [n], then the formula [(\forall x)F(x)] must be provable in P.

Now, if we meta-assume Hilbert’s \omega-rule for P, then it follows that, if P is consistent, then there is no P-formula [F(x)] for which, first, [\neg(\forall x)F(x)] is P-provable and, second, [F(n)] is P-provable for any given P-numeral [n].

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

D: The significance of \omega-consistency

To place the significance of \omega-consistency in a current perspective, we note that the standard model of the first order Peano Arithmetic PA (cf. Me64, p.107; Sc67, p.23, p.209; BBJ03, p.104) presumes [3] that the standard interpretation M of PA (under which the PA-formula [(\exists x)R(x)], which is merely an abbreviation for [\neg(\forall x)\neg R(x)] , interprets as true if, and only if, R(n) holds for some natural number n under M) is sound (cf. BBJ03, p.174).

Clearly, if such an interpretation of the existential quantifier is sound, it immediately implies that PA is necessarily \omega-consistent [4].

Since Brouwer’s main objection was to Hilbert’s presumption that such an interpretation of the existential quantifier is sound, Gödel explicitly avoided this assumption in his seminal 1931 paper (Go31, p.9) in order to ensure that his reasoning was acceptable as “constructive” and “intuitionistically unobjectionable” (Go31, p.26).

He chose, instead, to present the formal undecidability of his arithmetical proposition—and the consequences arising from it—as explicitly conditional on the assumption of the formal property of \omega-consistency for his Peano Arithmetic P under the unqualified—and, as we show below, mistaken—belief that:

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

E: Gödel: If the \omega -Rule is true, P cannot be completed

Now, Gödel’s significant achievement in Go31 was the discovery that, if P is consistent, then it was possible to construct a P-formula, [R(x)] [5], such that [R(n)] is P-provable for any given P-numeral [n] (Go31, p.25(2)), but [(\forall x)R(x)] is P-unprovable (Go31, p.25(1)).

However, it becomes apparent from his remarks in Go31 that Gödel considered his more significant achievement the further argument that, if P is assumed \omega -consistent, then both [(\forall x)R(x)] and [\neg (\forall x)R(x)] [6] are P-unprovable, and so P is incomplete!

This is the substance of Gödel’s Theorem VI (Go31, p.24).

Although this Theorem neither validated nor invalidated Hilbert’s \omega -rule, it did imply that assuming the rule led not to the completion of a Peano Arithmetic as desired by Hilbert, but to its essential incompletability!

F: The \omega -Rule is inconsistent with PA

Now, apparently, the possibility neither considered by Gödel in 1931, nor seriously since, is that a formal sytem of Peano Arithmetic—such as PA—may be consistent and \omega inconsistent.

If so, one would ascribe this omission to the `cover up’ factor mentioned by Feynman, since a significant consequence of Gödel’s reasoning—in the first half of his proof of his Theorem VI—is that it actually establishes PA as \omega inconsistent (as detailed in Corollary 9 of this preprint and Corollary 4 of this post).

In other words, we can logically show for Gödel’s formula [R(x)] that [\neg(\forall x) R(x)] is PA-provable, and that [R(n)] is PA-provable for any given PA-numeral [n].

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

G: Need: A paradigm shift in interpreting the quantifiers

Thus Gödel’s unqualified belief that:

PA is \omega-consistent

was misplaced, and Brouwer’s objection to Hilbert’s presumption—that the above interpretation of the existential quantifier is sound—was justified; since, if PA is consistent, then it is provably \omegainconsistent, from which it follows that the standard interpretation M of PA is not sound.

Hence we can no longer interpret `[\neg(\forall x)F(x)] is true’ maximally under the standard interpretation of PA as:

(i) The arithmetical relation F(n) is not always [7] true.

However, since the theorems of PA—when treated as Boolean functions—are Turing-computable as always true under a sound finitary interpretation \beta of PA, we can interpret `[\neg(\forall x)F(x)] is true’ minimally as:

(ii) The arithmetical relation F(n) is not Turing-computable as always true.

This interpretation allows us to conclude from Gödel’s meta-mathematical argument that we can construct a PA-formula [(\forall x)R(x)] that is unprovable in PA, but which is true under a sound interpretation of PA [8] although we may now no longer conclude from Gödel’s reasoning that there is an undecidable arithmetical PA-proposition.

Moreover, the interpretation admits an affirmative answer to Hilbert’s query: Is PA complete or completeable?

H: PA is algorithmically complete

In outline, the basis from which this conclusion follows formally is that:

(i) Gödel’s proof of the essential incompleteness of any formal system of Peano Arithmetic (Go31, Theorem VI, p.24) explicitly assumes that the arithmetic is \omega -consistent;

(ii) Rosser’s extension of Gödel’s proof of the essential incompleteness of any formal system of Peano Arithmetic (cf. Ro36, Theorem II, p.233) implicitly presumes that the Arithmetic is \omega -consistent (as detailed in this post);

(iii) PA is \omegainconsistent (as detailed in Corollary 9 of this preprint);

(iv) The classical `standard’ interpretation of PA (cf. Me64, section \S 2, pp.49-53; p107) over the structure [N]—defined as {N (the set of natural numbers); = (equality); ' (the successor function); + (the addition function); \ast (the product function); 0 (the null element)}— does not define a finitary model of PA (as detailed in the paper titled Evidence-Based Interpretations of PA presented at IACAP/AISB Turing 2012, Birmingham, UK in July 2012);

(v) We can define a sound interpretation \beta of PA—in terms of Turing-computability—which yields a finitary model of PA, but which does not admit a non-standard model for PA (as detailed in this paper);

(vi) PA is algorithmically complete in the sense that an arithmetical proposition F defines a Turing-machine TM_{F} which computes F as true under \beta if, and only if, the corresponding PA-formula [F] is PA-provable (as detailed in Section 8 of this preprint).

I: Gödel’s proof of his Theorem XI does not withstand scrutiny

Since Gödel’s proof of his Theorem XI (Go31, p.36)—in which he claims to show that the consistency of his formal system of Peano Arithmetic P can be expressed as a P-formula which is not provable in P—appeals critically to his Theorem VI, it follows that this proof cannot be applied to PA.

However, we show below that there are other, significant, reasons why Gödel’s reasoning in this proof must be treated as classically objectionable per se.

J: Why Gödel’s interpretation of the significance of his Theorem XI is classically objectionable

Now, in his Theorem XI, Gödel constructs a formula [W] [9] in P and assumes that [W] translates—under a sound interpretation of P—as an arithmetical proposition that is true if, and only if, a specified formula of P is unprovable in P.

Now, if there were such a P-formula, then, since an inconsistent system necessarily proves every well-formed formula of the system, it would follow that a proof sequence within P proves that P is consistent.

However, Gödel shows that his formula [W] is not P-provable (Go31, p.37).

He concludes that the consistency of any formal system of Peano Arithmetic is not provable within the Arithmetic. [10]

K: Defining meta-propositions of P arithmetically

Specifically, Gödel first shows how 46 meta-propositions of P can be defined by means of primitive recursive functions and relations (Go31, pp.17-22).

These include:

(\#23) A primitive recursive relation, Form(x), which is true if, and only if, x is the Gödel-number of a formula of P;

(\#45) A primitive recursive relation, xBy, which is true if, and only if, x is the Gödel-number of a proof sequence of P whose last formula has the Gödel-number y.

Gödel assures the constructive nature of the first 45 definitions by specifying (cf. Go31, p.17, footnote 34):

Everywhere in the following definitions where one of the expressions `\forall x‘, `\exists x‘, `\epsilon x (There is a unique x)’ occurs it is followed by a bound for x. This bound serves only to assure the recursive nature of the defined concept.

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

(\#46) A proposition, Bew(x), which is true if, and only if, (\exists y)yBx is true.

Thus Bew(x) is true if, and only if, x is the Gödel-number of a provable formula of P.

L: Expressing arithmetical functions and relations in P

Now, by Gödel’s Theorem VII (Go31, p.29), any recursive relation, say Q(x), can be represented in P by some, corresponding, arithmetical formula, say [R(x)], such that, for any natural number n:

If Q(n) is true, then [R(n)] is P-provable;

If Q(n) is false, then [\neg R(n)] is P-provable.

However, Gödel’s reasoning in the first half of his Theorem VI (Go31, p.25(1)) establishes that the above representation does not extend to the closure of a recursive relation, in the sense that we cannot assume:

If (\forall x)Q(x) is true (i.e, Q(n) is true for any given natural number), then [(\forall x)R(x)] is P-provable.

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

The reason: In recursive arithmetic, the expression `(\exists x)F(x)‘ is an abbreviation for the assertion:

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

In a formal Peano Arithmetic, however, the formula `[(\exists x)F(x)]’ is simply an abbreviation for `[\neg (\forall x)\neg F(x)]’, which, under a sound finitary interpretation of the Arithmetic can have the verifiable translation:

(**) The relation \neg F(x) is not Turing-computable as always true.

Moreover, Gödel’s Theorem VI establishes that we cannot conclude (*) from (**) without risking inconsistency.

Consequently, although a primitive recursive relation may be instantiationally equivalent to a sound interpretation of a P-formula, we cannot assume that the existential closure of the relation must have the same meaning as the interpretation of the existential closure of the corresponding P-formula.

However this, precisely, is the presumption made by Gödel in the proof of Theorem XI, from which he concludes that the consistency of P can be expressed in P, but is not P-provable.

M: Ambiguity in the interpreted `meaning’ of formal mathematical expressions

The ambiguity in the `meaning’ of formal mathematical expressions containing unrestricted universal and existential closure under an interpretation was emphasised by Wittgenstein (Wi56):

Do I understand the proposition “There is . . .” when I have no possibility of finding where it exists? And in so far as what I can do with the proposition is the criterion of understanding it … it is not clear in advance whether and to what extent I understand it.

N: Expressing “P is consistent” arithmetically

Specifically, Gödel defines the notion of “P is consistent” classically as follows:

P is consistent if, and only if, Wid(P) is true

where Wid(P) is defined as:

( \exists x) (Form(x) \wedge \neg Bew(x))

This translates as:

There is a natural number n which is the Gödel-number of a formula of P, and this formula is not P-provable.

Thus, Wid(P) is true if, and only if, P is consistent.

O: Gödel: “P is consistent” is always expressible in P

However, Gödel, then, presumes that:

(i) Wid(P) can be represented by some formula [W] of P such that “[W] is true” and “Wid(P) is true” are logically equivalent (i.e., have the same meaning) under a sound interpretation of P;

(ii) if the recursive relation, Q(x, p) (1931, p24(8.1)), is represented by the P-formula [R(x, p)], then the proposition “[(\forall x)R(x, p)] is true” is logically equivalent to (i.e., has the same meaning as) “(\forall x)Q(x, p) is true” under a sound interpretation of P.

P: The loophole in Gödel’s presumption

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

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

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

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

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

Bibliography

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

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

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

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

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

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

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

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

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

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

Notes

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

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

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

Return to 3: Following Hilbert.

Return to 4: Since we cannot, then, have that [\neg(\forall x)\neg R(x)] is PA-provable and that [\neg R(n)] is also PA-provable for any given numeral [n].

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

Return to 6: Gödel refers to these P-formulas only by their Gödel-numbers 17Gen \hspace{+.5ex} r and Neg(17Gen \hspace{+.5ex} r) respectively (cf. Go31, p.25, eqn.13).

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

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

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

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

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

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

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

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

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

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

“… There is an algorithm which, given any consistent set of axioms, will output a polynomial equation P = 0 which in fact has no integer solutions, but such that this fact can not be deduced from the given axioms. Here then is the true but unprovable Gödel sentence on which Penrose relies and in a particularly simple form at that. Note that the sentence is provided by an algorithm. If insight is involved, it must be in convincing oneself that the given axioms are indeed consistent, since otherwise we will have no reason to believe that the Gödel sentence is true”.

Note that the first part of Gödel’s argument in Theorem VI of his 1931 paper is that, if PA is consistent, then we can mechanically construct a PA formula—which, syntactically, is of the form [(\forall x)R(x)]—such that:

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

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

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

(iii) The PA-formula [(\forall x)R(x)] is formally unprovable in PA.

However, under standard interpretations of Alfred Tarski’s definitions of the satisfiability and truth of the formulas of a language L under an interpretation M, the L-formula [(\forall x)R(x)] is true in the interpretation M if, and only if, the interpreted relation R^{\prime}(x) is instantiationally satisfied in M (i. e. for any given element of M the interpreted relation can be ‘seen’ to hold in the interpretation).

If we take both L and M as PA (as detailed in ‘Evidence-Based Interpretations of PA‘), and take satisfiability in PA to mean instantiational provability in PA, we arrive at the formal definition of the truth of the PA-formula [(\forall x)R(x)] in PA as:

The PA-formula [(\forall x)R(x)] is formally true in PA if, and only if, the formula [R(x)] is provable in PA whenever we substitute a numeral [n] for the variable [x] in [R(x)].

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

(iv) The PA-formula [(\forall x)R(x)] is formally true in PA.

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

(v) The PA-formula [(\forall x)R(x)] is formally unprovable in PA, but formally true in PA.

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

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

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

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

(vi) The PA-formula [(\forall x)R(x)] is formally unprovable in PA, but intuitively true in the standard interpretation of PA.

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

(vii) The PA-formula [(\forall x)R(x)] is intuitively true in the standard interpretation of PA.

is both ambiguous and stronger than the meta-assertion:

(viii) The PA-formula [R(x)] is formally true in PA.

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

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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 x,\ P^{*}(x) does not hold’

usually denoted symbolically by ‘\neg(\forall x)\neg P^{*}(x)‘, we may always validly infer in the classical, Aristotlean, logic of predicates [1] that:

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

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

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 [\neg (\forall x) \neg F(x)] of a first order language S interprets as true under a sound interpretation of S, then we may always conclude that there must be some object s in the domain D of the interpretation such that, if the formula [F(x)] interprets as the unary relation F^{*}(x) in D, then the proposition F^{*}(s) 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 ‘[(\exists x)P(x)]‘ is an abbreviation for the formula ‘[\neg(\forall x)\neg P(x)]‘.

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

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

The significance of Aristotle’s particularisation for PA

In order to avoid intuitionistic objections to his reasoning, Kurt Gödel introduced the syntactic property of \omega-consistency [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 \omega-consistency explicitly was to avoid appealing to the semantic concept of classical arithmetical truth in Aristotle’s logic of predicates (which presumes Aristotle’s particularisation).

The two concepts are meta-mathematically equivalent in the sense that, if PA is consistent, then PA is \omega-consistent if, and only if, Aristotle’s particularisation holds under the standard interpretation of PA [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 \omega-consistent.

The structure \mathbb{N}

The structure of the natural numbers—namely:

N (the set of natural numbers);

= (equality);

S (the successor function);

+ (the addition function);

\ast (the product function);

0 (the null element).

The axioms of the first-order Peano Arithmetic PA

PA_{1}: [(x_{1} = x_{2}) \rightarrow ((x_{1} = x_{3}) \rightarrow (x_{2} = x_{3}))];

PA_{2}: [(x_{1} = x_{2}) \rightarrow (x_{1}^{\prime} = x_{2}^{\prime})];

PA_{3}: [0 \neq x_{1}^{\prime}];

PA_{4}: [(x_{1}^{\prime} = x_{2}^{\prime}) \rightarrow (x_{1} = x_{2})];

PA_{5}: [( x_{1} + 0) = x_{1}];

PA_{6}: [(x_{1} + x_{2}^{\prime}) = (x_{1} + x_{2})^{\prime}];

PA_{7}: [( x_{1} \star 0) = 0];

PA_{8}: [( x_{1} \star x_{2}^{\prime}) = ((x_{1} \star x_{2}) + x_{1})];

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

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

Generalisation in PA

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

Modus Ponens in PA

If [A] and [A \rightarrow B] are PA-provable, then so is [B].

The standard interpretation of PA

The standard interpretation \mathcal{I}_{PA(\mathbb{N},\ Standard)} of PA over the structure \mathbb{N} 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 [0] interprets as the integer 0;

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

(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 [F(x)] for which both [(\forall x)F(x)] and [\neg(\forall x)F(x)] are S-provable.

\omega-consistency

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

Soundness (formal system – non-standard)

A formal system S is sound under an interpretation \mathcal{I}_{S} with respect to a domain \mathbb{D} if, and only if, every theorem [T] of S translates as ‘[T] is true under \mathcal{I}_{S} in \mathbb{D}‘.

Soundness (interpretation – non-standard)

An interpretation \mathcal{I}_{S} of a formal system S is sound with respect to a domain \mathbb{D} if, and only if, S is sound under the interpretation \mathcal{I}_{S} over the domain \mathbb{D}.

Soundness in classical logic

In classical logic, a formal system S 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 L underlying any interpretation of S 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 L_{1} and L_{2} are implicitly assumed to be sound, S+L_{1} is sound, but S+L_{2} is not.

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

Algorithmic verifiability

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

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 \mathbb{N} if, and only if, they are algorithmically verifiable under the interpretation. [11]

Algorithmic computability

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

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 \mathbb{N} 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 EPR argument. Presented on 7’th April at the workshop on `Logical Quantum Structures‘ at UNILOG’2013, 4’th World Congress and School on Universal Logic, 29’th March 2013 – 7’th April 2013, Rio de Janeiro, Brazil.\

Notes

Return to 1: 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 \omega-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 \mathbb{N}.

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 \mathbb{N}; 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?.

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