Why resolving PvNP may require a paradigm shift

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Why resolving PvNP may require a paradigm

shift

The philosophical and mathematical significance of Aristotles

particularisation in the foundations of mathematics, logic and

computability

Bhupinder Singh Anand

Draft of November 23, 2009.

Abstract

I show why efforts which are founded on a classical logic that admits

Aristotles particularisation cannot resolve the PvNP Problem.

1 Introduction

In the September 2009 issue of the Communications of the Association of Com-puting Machinery, Vol. 52, #9, Lance Fortnow wrote1:

. . . we dont expect the P versus NP problem to be resolved in thenear future . . . None of us truly understand the P versus NP prob-lem, we have only begun to peel the layers around this increasinglycomplex question.

I shall seek to show whyas Fortnow appears to suggestresolving thePvNP problem may not be the major issue; the harder part may be altering ourattitudes and beliefs so that we can see what is obstructing such a resolution.

In particular, I shall show, first, why efforts which are founded on a classicallogic that admits Aristotles particularisation cannot resolve the PvNP problem;and, second, how the belief that Aristotles logic of predicates can be validlyapplied over the structure N2 of the natural numbers 0, 1, 2, . . . , has obstructed

efforts to prove that P=NP.

1.1 Why can we not immediately conclude from Godels

argument that P=NP?

For instance, I note that the standard definition of the class P due to Cook3

admits a number-theoretic function Fviewed formally as defining a uniquesubset L of the set of finite strings over some non-empty finite alphabet set

1[Fo09].2See Section 5 for the notation and definitions of technical terms as used in this paper.3[Cook].

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in P if, and only if, defines4 a deterministic Turing machine TM that

accepts L and runs in polynomial time.It follows that:

Lemma 1 A total arithmetical relationsayF(x)5is in P if, and only if, forany given natural numbern as input, the deterministic Turing machine TMF6

computes F(n) (when treated as a Boolean function) as either true or false inpolynomial time.

It further follows from Fortnows brief overview of Proof Complexity7 that:

Lemma 2 P=NP if we can define an arithmetical relation with a single variableF(x) over N such that:

(i) we can logically

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decide that, for any given natural number n,F(n) is true;9

(ii) there is no deterministic Turing machine TMF which can decidethat, given any natural numbern as input, F(n) is true.10 2

Now, in his seminal 1931 paper on formally undecidable propositions ofarithmetic, Kurt Godel showed how to construct a formulasay [R(x)]11inany Peano Arithmetic such that:

Lemma 3 If the first-order Peano Arithmetic PA is assumed to be consistent,then:

4The significance of making this distinction explicit is seen in Section 4, where I show whywe should not admit a number-theoretic function in P merely because it is instantiationally

equivalent to some member of P.5I shall use an asterisk to denote that the number-theoretical relation or function under

consideration is arithmetical in Godels sense; see [Go31], p.29.6I shall use this convention to denote the deterministic Turing machine defined by the

alphabet set F corresponding to F. Strictly speaking, it follows immediately from AlanTurings seminal 1936 paper on computable numbers [Tu36] that the quantifier-free stringsay [FQF]in the prenex normal form of any PA-formula [F] corresponds to an alphabet setPA which defines a deterministic Turing machine TMPA for computing the instantiationsof the arithmetical relation FQF over N, where F

QF is the interpretation of [FQF] under a

sound interpretation of PA.7See also [Cook]; [Ra02].8The distinction sought to be emphasised here is that a logical argumentsuch as G odels

reasoning in [Go31]may metamathematically ([Kl52], 14, p.55) establish the existence ofsome method for computing an arithmetical value by finitary means (in Hilberts sense [Hi25],p.378. See also [Kl52], 15, p.63; [Be59], p.233; [Wa63], p.41; [EC89], p.172.), without requiringthat any specific methodsuch as, say, a Turing machinebe necessary for computing thevalue.

9I note that F(n) is a tautology over N if, and only if, for any given natural number n,F(n) is true. The definition of a tautology only requires that the truth ofF(n) be decidableinstantiationally; it does not require that F(n) be partial recursive, and therefore decidablealgorithmically.

10This would be the case if, for instance, F(n) were a Halting type of function (such asthe one defined by Turing in [Tu36], p.132, 8); which, of course, implies that there is nodeterministic Turing machine TMF which can decide for any given natural number n as inputthat F(n) is true in polynomial time.

11In his paper, Godel refers to this formula only by its Godel-number r ([Go31], p.25,eqn.12).

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(i) for any numeral [n ], the PA-formula [R(n)] is provable in PA12;

(ii) the PA-formula [R(x)] is not provable in PA13. 2

If we denote the interpretation of [R(x)] under an interpretation of the Arith-metic over N by R(x)14, it follows that, if PA has a sound interpretation, thenwe can logically conclude that, for any given natural number n, the arithmeticalproposition R(n) is true.

The question arises:

Query 1 Assuming that PA has a sound interpretation over N15, why can wenot immediately conclude from Godels argument that, if the PA-formula [R(x)]is not provable in PA, then there is no deterministic Turing machine TMRwhich can decide that, given any natural numbern as input, R(n) is true (andtherefore P=NP)?

1.2 The significance of-consistency for the PvNP prob-

lem

I shall show how the answer to this query is obscured by the persisting beliefthat Aristotles 2000-year old logic of predicates remains valid even when appliedover an infinite domain such as N16.

A critical component of this belief is Aristotles particularisation, which isthe postulation that:

Aristotlean Postulate 1 From the assertion:

It is (logically17) not the case that, for any given natural numbern,F(n) does not hold

we may always conclude:

There is some (unspecified) natural number m such that F(m)holds.18

Now, in his 1931 paper, Godel introduced the concept19:

Definition 1 PA is -consistent if, and only if, there is no PA-formula [F(x)]such that:

(i) for any given PA-numeral [n ], the PA-formula [F(n) ] is PA-

provable;(ii) the PA-formula [(x)F(x)] is provable in PA.

12See [Go31], p.26(2).13This follows by Generalisation from Godels actual argument (see [Go31], p.25(1)), which

is that the arithmetical formula [(x)R(x)]to which Godel refers only by its Godel-number17Gen r ([Go31], p.25, eqn.13)is not provable in any Peano Arithmetic.

14I shall aim to use this notation consistently in this paper.15I define such an interpretation finitarily in Section 6; and show that it is sound in Section

7; see also [An09c].16Despite the indisputable objections raised by L. E. J. Brouwer in 1908 [Br08].17I shall presume this qualification to be implicit henceforth.18Clearly, the converse is always true.19[Go31], p.23.

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It follows that:

Lemma 4 If PA is consistent but not -consistent, then there is some PA-formula [F(x)] such that, under any sound interpretationsay IP A(Sound)ofPA:

(i) for any given numeral [n ], the PA-formula [F(n)] interprets astrue under IP A(Sound);

(ii) the PA-formula [(x)F(x)] interprets as true underIP A(Sound). 2

Now, by Alfred Tarskis standard definitions of the satisfaction, and truth,of the formulas of a formal system such as PA under an interpretation 20, say

IP A(N), over N:

Lemma 5 If the PA-formula [(x)F(x) ] interprets as true under IP A(N),then it is not the case that, for any given PA-numeral [n ], the PA-formula[F(n)] interprets as true under IP A(N). 2

If the interpretation IP A(N) admits Aristotlean particularisation over N21,

it follows that:

Aristotlean Postulate 2 If the PA-formula [(x)F(x) ] interprets as trueunder IP A(N), then there is some unspecified PA-numeral [m ] such that thePA-formula [F(m)] interprets as false under IP A(N). 2

So, the belief that Aristotles particularisation holds over N yields the pos-tulate:

Aristotlean Postulate 3 If PA is consistent, there can be no PA-formula[F(x) ] such that, under any sound interpretation, sayIP A(N,Sound), of PA over

N:

(i) for any given numeral [n ], the PA-formula [F(n)] interprets astrue under IP A(N,Sound);

(ii) the PA-formula [(x)F(x)] interprets as true underIP A(N,Sound).

Hence, the belief that Aristotles particularisation is valid over N implies22

that:

Aristotlean Postulate 4 If PA is consistent, then it is -consistent.

Clearly, if PA is -consistent then, since [n = n] is PA-provable for any givenPA-numeral [n], we cannot have that [(x)(x = x)] is PA-provable. Thus,since an inconsistent PA proves [(x)(x = x)], an -consistent PA cannot beinconsistent. It follows that:

20[Ta33]; see also [Ho01] for an explanatory exposition. However, for standardisation andconvenience of expression, we follow the formal exposition of Tarskis definitions given in[Me64], p.50.

21As, for instance, in [Me64], pp.51-52 V(ii).22The above argument is made explicit in view of Davis remark in [Da82], p.129, that such

a proof of-consistency may be . . . open to the objection ofcircularity.

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Aristotlean Postulate 5 PA is consistent if, and only if, it is -consistent.23

2 The first paradigm shift: Godels reasoning

implies that Aristotles particularisation does

not hold over the natural numbers

Now, the first component of the paradigm shift24 lies in recognising thatcontrary to accepted dogma25Godels reasoning in Theorem VI26 of his 1931paper on formally undecidable arithmetical propositions admits the conclusionthat (see Section 10)27:

Lemma 6 If PA is consistent, then it is not-consistent. 2

It follows that:

Lemma 7 Aristotles particularisation does not hold over N. 2

As the classical, standard, interpretation of PAsay IP A(Standard/Tarski)28

appeals to Aristotles particularisation29, this paradigm shift also involvesrecognisingagain, contrary to accepted dogmathat the interpretation IP A(Standard/Tarski) is not sound, and does not yield a model of PA

30.Now, formal quantification in computational theory is currently interpreted

as in classical logic31so as to admit Aristotlean particularisation over N asaxiomatic32.

However, if Aristotles particularisation does not hold over N, it would ex-

plain to some extent why efforts to resolve the PvNP problem by argumentsthat appeal to classical Aristotlean logic cannot prevail.

23It is pertinent to note thatin order to avoid intuitionistic objections to his reasoningin [Go31]Godel explained at some length (in his introduction on p.9 of [Go31]) that hisreasons for introducing the syntactic property of -consistency as an explicit assumption inhis formal reasoning (see [Go31], p.23 and p.28), was to avoid appealing to the stronger,semantic, concept of classical arithmetical trutha concept which is implicitly based on anintuitionistically objectionable logic that assumes Aristotlean particularisation is valid overN, and under whichas shown herePA is consistent if, and only if, it is -consistent.Moreoveras I show in [An09b]Rossers extension of Godels argument ([Ro36]) succeedsin avoiding an explicit assumption of-consistency only by implicitly appealing to Aristotleanparticularisation.

24In Kuhns sense; see [Ku62], p.52.25See, for instance, Davis remarks in [Da82], p.129(iii) that . . . there is no equivocation.

Either an adequate arithmetical logic is -inconsistent (in which case it is possible to prove

false statements within it) or it has an unsolvable decision problem and is subject to thelimitations of Godels incompleteness theorem.

26[Go31], p.24-26.27See also [An09a].28See Section 5.29See, for instance, [Me64], p.107 and p.52(V)(ii).30I note that finitists of all huesranging from Brouwer [Br08] to Alexander Yessenin-

Volpin [He04]have persistently questioned the soundness of the standard interpretationIPA(Standard/Tarski) of PA.

31See [Hi25], p.382; [HA28], p.48; [Be59], pp.178 & 218.32In the sense of being intuitively obvious. See, for instance, [Da82], p.xxiv; [Rg87], p.308

(1)-(4); [EC89], p.174 (4); [BBJ03], p.102.

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3 The second paradigm shift: Turings reason-

ing implies that arithmetical satisfiability canbe defined algorithmically

The second component of the paradigm shift lies in recognising that Turings1936 paper on computable numbers admits a sound, finitary, interpretation ofPAsay IP A(rouwer/ Turing)in which arithmetical satisfiability is definablein terms of Turing-computability.33.

Specifically, this interpretation (see Section 634) differs from the standardinterpretation IP A(Standard/Tarski) of PA only in the requirement that, whereasthe latter interpretation defines the satisfaction of atomic PA-formulas in termsof a subjective, and essentially unverifiable, intuitive decidability35, the for-mer defines the satisfaction of atomic PA-formulas under IP A(rouwer/Turing )

in terms of an objective, and instantiationally verifiable, algorithmic decidabil-ity36.

The comparative definitions of the satisfiability of the atomic formulas ofPA under the two interpretations are as follows:

Definition (Standard/Tarskian satisfiability) If [A(t1, . . . , tn)] is an atomic well-formed formula of PA, A(t1, . . . , t

n) the corresponding interpretation over N,

and the sequence of PA-terms [a1], . . . , [an] interprets over N as a1 . . . , a

n,

then a1 . . . , an satisfies [A(t1, . . . , tn)] under IP A(Standard/Tarski) if, and only

if, A(a1, . . . , an) holds over N;

Definition (Brouwer/Turing satisfiability): If [A(t1, . . . , tn)] is an atomic well-formed formula of PA, A(t1, . . . , t

n) the corresponding interpretation over N,

and the sequence of PA-terms [a1], . . . , [an] interprets over N as a1 . . . , a

n,

then a1 . . . , an satisfies [A(t1, . . . , tn)] under IP A(rouwer /Turing) if, and only if,A(a1, . . . , a

n) is TMA-computable (when treated as a Boolean expression) as

holding over N, where TMA is the Turing machine defined by A(t1, . . . , tn)

37.

The satisfaction, and truth, of compound formulas of PA under IP A(rouwer/Turing) follow Tarskis standard, inductive, definitions (see Section 6

38) asusual39. They yield the following consequences (see Section 740):

33It immediately follows from this thatas suspected, and sought, by David Hilbert [Hi27]PA is consistent (see Section 7; see also [An09c]).

34See also [An09c].35Note that Tarski defines the formal sentence P as True if and only if pwhere p is the

proposition expressed by P (see [Ta33]). In other words, the sentence Snow is white is Trueif, and only if, it is subjectively true in all cases; and it is subjectively true in a particularcase if, and only if, it expresses the subjectively verifiable fact that snow is white in thatparticular case. Thus, for Tarski the commonality of the satisfaction of the atomic formulasof a language under an interpretation (cf. [Me64], p.51(i)) is axiomatic, and need not beobjectively verifiable.

36Moreover, it follows that introduction of an algorithmic method is necessary for the de-cidability of the satisfaction and truth of PA-formulas under a sound interpretation of PA (seeSection 9; see also [An09c]).

37It follows from Turings seminal 1936 paper ([Tu36], pp.138-139) that every quantifier-freearithmetical relation F(x) (when interpreted as a Boolean function) defines a Turing machineTMF such that F(x) is TMF-computable if, and only if, for any given natural number n,TMF will compute F(n) as either true, or as false, over the structure N.

38See also [An09c].39See, for instance, [Me64], pp.49-51.40See also [An09c].

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Lemma The PA-axioms PA1 to PA841 interpret as tautologies under the inter-

pretation IP A(rouwer/Turing).2

Lemma For any given PA-formula [F(x) ], the Induction axiom schema PA9interprets under IP A(rouwer/Turing ) as a tautology. 2

Lemma Generalisation preserves truth under IP A(rouwer/Turing). 2

Lemma Modus Ponens preserves truth under IP A(rouwer/T uring). 2

Hence the interpretation IP A(rouwer/Turing) of PA is sound, and defines afinitary model of PA.42

3.1 A case for P=NP

Moreover, it follows from the above that (see Section 843):

Provability Theorem for PA A total arithmetical relation F

(x)whentreated as a Boolean functiondefines a deterministic Turing machine TMFwhich computes F(x) as always true (i.e., true for any given natural numberinput n when quantification in F(x) is interpreted Turing-computably under

IP A(rouwer/Turing) as indicated above) if, and only if, the PA-formula [F(x)]is provable in PA. 2

It follows from the Provability Theorem for PA that:

Theorem 1 If an arithmetical relationF is decidable by the deterministic Tur-ing machine TMF as a tautology metamathematically, then the formula [F] isPA-provable. 2

It further follows that:

Lemma 8 For any arithmetical relationF(x):

If the PA-formula [F(x)] is not provable in PA, then there is no de-terministic Turing machine TMF such that, given any natural num-bern as input, TMF decides thatF(n) is true. 2

Hence:

Corollary 1 Godels relationR(x) is such that:

(i) for any given natural number n, R(n) is necessarily true;

(ii) there is no deterministic Turing machine TMR which can showthat, for any given natural numbern as input, R(n) is true. 2

This is the assertion that Godels arithmetical relation R(x) is metamath-ematically decidable as always true, but not algorithmically decidable by thedeterministic Turing machine TMR as always true. It follows that:

Theorem 2 P=NP. 2

41See Section 5 for a formal statement of the PA axioms.42We thus have a finitary proof that PA is consistent.43See also [An09c].

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4 The third paradigm shift: Churchs Thesis is

false

The third component of the paradigm shiftand possibly the most signifi-cant one for appreciating why efforts to prove P=NP remain unsuccessfulliesin recognising thatcontrary to accepted dogma44the term effective com-putability can be precisely defined.

For instance, we can define:

Definition 2 A computation is effectively computable if, and only if, it halts infinite time using finite resources.

Definition 3 A number theoretic function F is effectively computable instanti-ationally if, and only if, given any sequence of natural number values for its free

variables (assumed finite), F is effectively computable.

Definition 4 A number-theoretic function is effectively computable if, and onlyif, it is effectively computable instantiationally.

In other words, a number-theoretic function such as F(x) is effectively com-putable if, and only if, for any given natural number n, F(n) is effectivelycomputable .

Prima facie, these definitions adequately capture our intuitive understandingof the term effective computability.

Now, classical theory argues that:

Lemma 9 Every Turing-computable function (or relation, treated as a Booleanfunction) F is partial recursive , and, ifF is total , thenF is recursive45. 2

Lemma 10 Every partial recursive function (or relation, treated as a Booleanfunction) is Turing-computable46. 2

It follows that the followingessentially unverifiable but refutablethesesare classically equivalent47:

Standard Churchs Thesis48 A number-theoretic function (or re-lation, treated as a Boolean function) is effectively computable if,and only if, it is partial-recursive49.

Standard Turings Thesis50 A number-theoretic function (or re-lation, treated as a Boolean function) is effectively computable if,and only if, it is Turing-computable51.

44See Section 10; also [Kl52], p.300; [Me64], p.227; [Rg87], p.20; [EC89], p.85; [BBJ03],p23.

45cf. [Me64], p.233, Corollary 5.13.46cf. [Me64], p.237, Corollary 5.15.47cf. [Me64], p.237.48Churchs (original) Thesis The effectively computable number-theoretic functions are the

algorithmically computable number-theoretic functions [Ch36].49cf. [Me64], p.227.50After describing what he meant by computable numbers in the opening sentence of his

seminal paper on Computable Numbers [Tu36], Turing immediately expressed this thesisalbeit informallyas: . . . the computable numbers include all numbers which could naturallybe regarded as computable.

51cf. [BBJ03], p.33.

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4.1 The Church and Turing Theses imply P=NP

However, we note that, in its classical formulations, Churchs Thesis ignores thedoctrine of Occams razor by postulating a strong identityand not simply aweak equivalencebetween an effectively computable number-theoretic functionand some algorithmically computable function (see Section 11).

Consequently, Churchs Thesis (Turings Thesis) does not admit the possibil-ity of an arithmetical function F that is effectively computable instantiationally(i.e., constructively computable), but not partial recursive (since it is not com-putable by the Turing machine TMF).

Now a consequence of the Provability Theorem for PA is that:

Theorem 3 (First Tautology Theorem) Godels tautology R(n) is effectivelycomputable as true for any given natural number n.

Proof Godel has defined a primitive recursive relation, xBP Ay that holds if,and only if, y is the Godel-number of a PA-formula, say [Y], and x the Godel-number of a PA-proof of [Y] ([Go31], p22, dfn. 45).

Since every primitive recursive relation is Turing-computable (when treatedas a Boolean function), xBP Ay defines a deterministic Turing machine TMBthat halts on any natural number values ofx and y.

Now, Godel has shown, by a meta-mathematical argument52, that we canconstruct a PA-formula [(x)R(x)] which is not PA-provable, but is such that[R(n)] is PA-provable for any given numeral [n].

Hence, ifg[R(1)], g[R(2)], . . . are the Godel-numbers of the PA-formulas [R(1)],[R(2)], . . . , it follows that, for any given natural number n, when the naturalnumber value g[R(n)] is input for y, the deterministic Turing machine TMB must

halt for some value ofx, which is the Godel-number of some PA-proof of [R(n)].Thus, R(n) is effectively computable as true for any given natural number

n. 2

It now follows from the First Tautology Theorem that, since there is anumber-theoretic relation whichtreated as a Boolean functionis effectivelycomputable but not partial recursive53:

Corollary 2 The Church and Turing theses do not hold. 2

The significance of the Church and Turing theses for the PvNP problem isthatas noted in Section 1.1P=NP if there is an arithmetical tautology R(x)which is not decidable by the deterministic Turing machine TMR as a tautology.

It thus follows from the First Tautology Theorem that:

Lemma 11 If P=NP, then the Church and Turing theses are false. 2

Corollary 3 If the Church and Turing theses are true, then P=NP. 2

Since belief in the validity of the Church and Turing theses seems widespreadamongst computationalists, it would explain why the obstacles to proving thatP=NP appear unsurmountable.

52[Go31], p25(1).53Since the axioms of a set theory such as ZF do not admit such a distinction, the above

also suggests that set-theoretic arguments may not be able to resolve the PvNP problem.

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5 Appendix A: Notation, Definitions and Com-

ments

Aristotlean particularisation This holds that an assertion such as, Thereexists an unspecified x such that F(x) holdsusually denoted symbolically by(x)F(x)can always be validly inferred in the classical, Aristotlean, logic ofpredicates54 from the assertion, It is not the case that, for any given x, F(x)does not holdusually denoted symbolically by (x)F(x).

Notation In this paper I use square brackets to indicate that the contentsrepresent a symbol or a formula of a formal theory, generally assumed to bewell-formed unless otherwise indicated by the context.

In other words, expressions inside the square brackets are to be only viewedsyntactically as juxtaposition of symbols that are to be formed and manip-

ulated upon strictly in accordance with specific rules for such formation andmanipulationin the manner of a mechanical or electronic devicewithout anyregards to what the symbolism might represent semantically under an interpre-tation that gives them meaning.

Moreover, even though the formula [F(x)] of a formal Arithmetic may inter-pret as the arithmetical relation expressed by F(x), the formula [(x)R(x)]need not interpret as the arithmetical proposition denoted by the abbrevia-tion (x)R(x). The latter denotes the phrase There is some x such thatR(x). As Brouwer had noted55, this concept is not always capable of an un-ambiguous meaning that can be represented in a formal language by the formula[(x)R(x)].

By expressed I mean here that the symbolism is simply a short-hand abbrevi-ation for referring to abstract concepts that may, or may not, be capable of aprecise meaning. Amongst these are symbolic abbreviations which are intendedto express the abstract conceptsparticularly those of existenceinvolved in

propositions that refer to non-terminating processes and infinite aggregates.

Provability A formula [F] of a formal system S is provable in S (S-provable) if,and only if, there is a finite sequence of S-formulas [F1], [F2], ..., [Fn] such that[Fn] is [F] and, for all 1 i n, [Fi] is either an axiom of S or a consequenceof the axioms of S, and the formulas preceding it in the sequence, by means ofthe rules of deduction of S.

The structure N The structure of the natural numbersnamely, {N (the setof natural numbers); = (equality); (the successor function); + (the additionfunction); (the product function); 0 (the null element)}.

The axioms of first-order Peano Arithmetic (PA)

PA1 [(x1 = x2) ((x1 = x3) (x2 = x3))];

PA2 [(x1 = x2) (x1 = x2)];PA3 [0 = x

1];

PA4 [(x1 = x

2) (x1 = x2)];

PA5 [(x1 + 0) = x1];PA6 [(x1 + x

2) = (x1 + x2)

];PA7 [(x1 0) = 0];PA8 [(x1 x

2) = ((x1 x2) + x1)];

PA9 For any well-formed formula [F(x)] of PA:[(F(0) (x)(F(x) F(x))) (x)F(x)].

54[HA28], pp.58-59.55[Br08]; see also [An08].

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

Modus Ponens in PA If [A] and [A B] are PA-provable, then so is [B].Standard interpretation of PA The standard interpretation IP A(Standard/Tarski) of PA over the structure N is the one in which the logical constants havetheir usual interpretations56 in Aristotles logic of predicates, and57:

(a) the set of non-negative integers is the domain;(b) the integer 0 is the interpretation of the symbol [0];(c) the successor operation (addition of 1) is the interpretation of the [ ]

function;(d) ordinary addition and multiplication are the interpretations of [+] and

[];(e) the interpretation of the predicate letter [=] is 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 [(x)F(x)] and [(x)F(x)] areS-provable.

-consistency A formal system S is -consistent if, and only if, there is noS-formula [F(x)] for which, first, [(x)F(x)] is S-provable and, second, [F(a)]is S-provable for any given S-term [a].

Soundness (formal system) A formal system S is sound under an interpre-tation IS if, and only if, every theorem [T] of S translates as [T] is true under

IS .

Soundness (interpretation) An interpretation IS of a formal system S issound if, and only if, S is sound under the interpretation IS .

Soundness in classical logic. In classical logic, a formal system S is sometimesdefined as sound if, and only if, it has an interpretation; and an interpretationis defined as the assignment of meanings to the symbols, and truth-values to thesentences, of the formal system. Moreover, any such interpretation is a model ofthe formal system. This definition suffers, however, from an implicit circularity: theformal 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 definedsoundness of a formal system under an interpretation from the implicit soundnessof the formal logic underlying the interpretation. This admits the case where, evenif L1 and L2 are implicitly assumed to be sound, S + L1 is sound, but S + L2 is not.Moreover, an interpretation of S is now a model for S if, and only if, it is sound.58

Categoricity A formal system S is categorical if, and only if, it has a soundinterpretation and any two sound interpretations of S are isomorphic.59

6 Appendix B: A finitary interpretation of PAIn the interpretation IP A(rouwer/T uring) of PA, we interpret the non-logicalconstants as in the standard interpretation IP A(Standard/Tarski) of PA in theusual manner, but interpret the logical constants algorithmically.

Specifically, the interpretation IP A(rouwer/Turing) of PA is obtained if, inTarskis inductive definitions of the satisfaction and truth of the formulas of a

56See [Me64], p.49.57See [Me64], p.107.58My thanks to Professor Rohit Parikh for highlighting the need for making such a distinc-

tion explicit.59Compare [Me64], p.91.

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formal system such as PA under the standard interpretation IP A(Standard/Tarski)

of PA(1a) and (2)-(6) below60

we apply Occams razor and weaken (1a) byreplacing it with the algorithmic definition of Turing-satisfiability (1b).

6.1 Defining satisfiability

Thus, the comparative definitions of the satisfiability of the atomic formulas ofPA under the two interpretations are as follows:

Definition 1a (Standard/Tarskian satisfiability) If [A(t1, . . . , tn)] is an atomicwell-formed formula of PA, A(t1, . . . , t

n) the corresponding interpretation over

N, and the sequence of PA-terms [a1], . . . , [an] interprets over N as a1 . . . , a

n,

then a1 . . . , an satisfies [A(t1, . . . , tn)] under IP A(Standard/Tarski) if, and only if,

A(a1, . . . , an) holds over N;

Definition 1b (Brouwer/Turing satisfiability) If [A(t1, . . . , tn)] is an atomicwell-formed formula of PA, A(t1, . . . , t

n) the corresponding interpretation over

N, and the sequence of PA-terms [a1], . . . , [an] interprets over N as a1 . . . , an,

then a1 . . . , an satisfies [A(t1, . . . , tn)] under IP A(rouwer /Turing) if, and only if,

A(a1, . . . , an) is TMA -computable (when treated as a Boolean expression) as

holding over N, where TMA is the Turing machine defined by A(t1, . . . , tn)

61.

6.2 Defining truth under an interpretation

However, both IP A(Standard/Tarski) and IP A(rouwer/T uring) identically definethe satisfaction and truth of the compound formulas of PA inductively as usualunder the corresponding interpretation as follows:

Definition 2 A sequence s satisfies [A] under the interpretation if, and onlyif, s does not satisfy [A];

Definition 3 A sequence s satisfies [A B] under the interpretation if, andonly if, either s does not satisfy [A], or s satisfies [B];

Definition 4 A sequence s satisfies [(xi)A] under the interpretation if, andonly if, every denumerable sequence over N which differs from s in at most theith component satisfies [A];

Definition 5 A well-formed formula [A] of PA is true under the interpretationif, and only if, every denumerable sequence over N satisfies [A];

Definition 6 A well-formed formula [A] of PA is false under the interpretationif, and only if, no sequence over N satisfies [A].

60cf. [Me64], pp.50-52.61It follows from Turings seminal 1936 paper ([Tu36], pp.138-139) that every quantifier-

free arithmetical relation F(x) (when interpreted as a Boolean function) defines a Turingmachine TMF (in the general case, TMF is defined by the quantifier-free expression in theprenex normal form ofF(x)) such that F(x) is TMF-computable if, and only if, for any givennatural number n, TMF will compute F(n) as either true, or as false, over the structure N.

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7 Appendix C: The finitary interpretation

IPA(rouwer/Turing) of PA is soundSince the PA-axioms PA1 to PA8 do not involve any quantification, it followsstraightforwardly from Alan Turings 1936 seminal paper on computable num-bers62 that:

Lemma 12 The PA-axioms PA1 to PA8 interpret as tautologies under the in-terpretation IP A(rouwer/T uring). 2

Further:

Lemma 13 For any given PA-formula [F(x) ], the Induction axiom schemaPA9 interprets under IP A(rouwer/Turing) as a tautology.

Proof The two meta-assertions:

[F(0) (x)(F(x) F(x))] is true under IP A(rouwer/Turing)

and

[(x)F(x)] is true under IP A(rouwer/Turing)

both mean63:

For any given natural number n, the Turing machine TMF com-putes F(n) as true

where F(x) is the interpretation of [F(x)] under IP A(rouwer/Turing )

. 2

Similarly:

Lemma 14 Generalisation interprets underIP A(rouwer/Turing ) as a tautology.

Proof The two meta-assertions:

[F(x)] is true (i.e., satisfied for any given x) under IP A(rouwer/Turing)

and

[(x)F(x)] is true under IP A(rouwer/Turing)

both mean:

For any given natural number n, the Turing machine TMF com-putes F(n) as true

where F(x) is the interpretation of [F(x)] under IP A(rouwer/Turing ). 2

It is also straightforward to see that:

62[Tu36].63I note that the interpretation IPA(rouwer/Turing) settles the Poincare-Hilbert debate

(see [Hi27], p.472; also [Br13], p.59; [We27], p482; [Pa71], p.502-503.) in the latters favour.Poincare believed that the Induction Axiom could not be justified finitarily, as any suchargument would necessarily need to appeal to infinite induction. Hilbert believed that afinitary proof of the consistency of PA was possible.

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Lemma 15 Modus Ponens preserves truth under IP A(rouwer/Turing). 2

We thus have that:

Lemma 16 The axioms of PA are constructively satisfied/true under the fini-tary interpretation IP A(rouwer/Turing), and the rules of inference of PA pre-serve the properties of satisfaction/truth under IP A(rouwer/Turing). 2

It follows that the finitary interpretation IP A(rouwer/Turing) of PA is sound,and so it defines a finitary model of PA.

8 Appendix D: A Provability Theorem for PA

I show that PA is categorical, and can have no non-standard model, since it is

algorithmically complete in the sense that:

Theorem 4 (Provability Theorem for PA) A total arithmetical relationF(x)when treated as a Boolean functiondefines a deterministic Turing machineTMF which computes F

(x) as always true (i.e., true for any given naturalnumber input n) if, and only if, the corresponding PA-formula [F(x) ] is PA-provable.

Proof It follows from Turings seminal 1936 paper 64 that every quantifier-freearithmetical relation F(x) (when interpreted as a Boolean function) defines adeterministic Turing machine TMF such that F

(x) is TMF-computable if, andonly if, for any given natural number n, TMF will compute F

(n) as eithertrue, or as false, over the structure N.

In the general case, TMF is defined by the quantifier-free expression in theprenex normal form of F(x), and any quantification in F(x) is to be inter-preted Turing-computably under IP A(rouwer/T uring) as detailed above.

Now, we have that:

(a) [(x)F(x)] is defined as true under the interpretation IP A(rouwer/Turing)if, and only if, the Turing machine TMF computes F

(n) as always true(i.e., as true for any given natural number n) under IP A(rouwer/Turing );

(b) [(x)F(x)] is an abbreviation of [(x)F(x)], and is defined as trueunder IP A(rouwer/Turing) if, and only if, it is not the case that the Turi-ng-machine TMF computes F

(n) as always false (i.e., as false for anygiven natural number n) under IP A(rouwer/Turing).

Moreover, since IP A(rouwer/Turing) is sound, it defines a finitary model ofPA over Nsay MP A()such that:

If [(x)F(x)] is PA-provable, then the Turing machine TMF computes F(n)as always true (i.e., as true for any given natural number n) in MP A();

If [(x)F(x)] is PA-provable, then it is not the case that the Turing machineTMF computes F

(n) as always true (i.e., as true for any given natural numbern) in MP A().

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

64[Tu36].

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(i) There is a finitary modelsay M1 of PA+[(x)F(x)] in which the

Turing machine TMF computes F

(n) as always true (i.e., as true forany given natural number n).(ii) There is a finitary modelsay M2 of PA+[(x)F(x)] in which it is

not the case that the Turing machine TMF computes F(n) as always

true (i.e., as true for any given natural number n).

Hence PA is algorithmically complete, in the sense that a total arithmeticalrelation F(x)when treated as a Boolean functiondefines a Turing machineTMF which computes F

(x) as always true if, and only if, the correspondingPA-formula [F(x)] is PA-provable65. 2

It follows that PA can have no non-standard model, and so we have that:

Corollary 4 PA is categorical. 2

9 Appendix E: Introduction of an algorithmic

method is necessary for the decidability of the

satisfaction and truth of PA-formulas under a

sound interpretation of PA

The question arises: Is the introduction of an algorithmic method for the decid-ability of the satisfaction and truth of PA-formulas under a sound interretationof PA necessary?

I give an affirmative answer by showing that, even if the atomic PA-formulasare defined verifiably under an interpretation, say IP A(Godel), the interpretationis not sound 66 if it admits Aristotles particularisation.

Thus, I define the satisfaction of the atomic PA-formulas under IP A(Godel)as follows:

Definition 1c (Godelian satisfiability) If [A(t1, . . . , tn)] is an atomic well-formedformula of PA then the sequence of PA-terms [a1], . . . , [an] satisfies [A(t1, . . . , tn)]under IP A(Godel) if, and only if, [A(a1, . . . , an)] is PA-provable.

The satisfaction and truth of the compound formulas of PA are definedinductively under the interpretation IP A(Godel) as usual by definitions (2)-(6)in Section 6 above.

If we accept that (1c) is a consistent definition of satisfiability, then it isstraightforward to show that IP A(Godel) is sound.

However, it also follows from the definition (1c) above and definitions (2)-(6)in Section 6 that:

Lemma 17 (Godelian universality) A PA-formula such as [(x)A(x)] inter-prets as true under IP A(Godel) if, and only if, for any given numeral [n], [A(n)]is PA-provable.

Lemma 18 (Godelian particularisation) A PA-formula such as [(x)A(x)]the abbreviation for [(x)A(x)]interprets as true under IP A(Godel) if, andonly if, it is not true that for any given numeral [n], [A(n)] is PA-provable.

65Note that [(x)F(x)] is PA-provable if, and only if, [F(x)] is PA-provable.66Compare [Br08]; see also [Br23], p.336; [Br27], p.491; [We27], p.483.

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Further, it folows from the Provability Theorem for PA in Section 8 above

that the Godelian PA-formula [R(x)] is such that:(i) [(x)R(x)] is not PA-provable;

(ii) [(x)R(x)] is PA-provable;

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

Now, if IP A(Godel) is sound, then (ii) implies that it is not the case that, forany given numeral [n], [R(n)] is PA-provable.

It follows that IP A(Godel) is not sound.We conclude that:

Theorem 5 If an interpretation IP A(S) of PA lacks specification of an algo-rithmic method for determining the satisfaction of the atomic formulas of PA,

then this lack is reflected in the non-constructivity of the interpretations of theuniversal and existential quantifiers of PA under IP A(S).

Corollary 5 If an interpretation IP A(S) of PA lacks specification of an algo-rithmic method for determining the satisfaction of the atomic formulas of PA,then IP A(S) cannot distinguish between arithmetical relations that are algorith-mically decidable, and those that are instantiationally, but not algorithmically,decidable.

10 Appendix F: A consistent PA is not omega-

consistent

In his 1931 paper67, Godel defined a Peano Arithmetic P and showed that:

Lemma 19 If P is consistent and [(x)R(x)] is assumed P-provable, then [(x)R(x) ] is P-provable68. 2

By Godels definition of P-provability69, it follows that:

Lemma 20 There is a finite sequence [F1], . . . , [Fn ] of P-formulas such that[F1 ] is [(x)R(x) ], [Fn ] is [(x)R(x) ], and, for 2 i n, [Fi ] is either aP-axiom or a formal consequence of the preceding formulas in the sequence bythe rules of inference of P. 2

Now:

Lemma 21 Under every sound interpretation of P, the sequence [F1], . . . , [Fn]of P-formulas interprets as a finite sequence F1 , . . . , F

n of arithmetical proposi-

tions such thatF1 is the interpretation of [(x)R(x)], Fn is the interpretation

of [(x)R(x) ], and, for 2 i n, Fi is either the interpretation of a P-axiom, or a logical consequence of the preceding formulas in the sequence by theinterpretation of the rules of inference of P. 2

We thus have that:

67[Go31].68This follows from Godels argument in [Go31], p.26(1).69[Go31], p.22, Definitions #44-46.

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Lemma 22 There is no sound interpretation of P under which [(x)R(x)] in-

terprets as true and [(x)R(x) ] as false.2

Since both [(x)R(x)] and [(x)R(x)] are closed P-formulas, it, further,follows that:

Lemma 23 The formula [(x)R(x) (x)R(x) ] interprets as true underevery sound interpretation of P. 2

In other words, the implication IfF1 then F

n holds in any sound interpre-tation of P. By Godels completeness theorem, it follows that:

Lemma 24 ((x)R(x) (x)R(x)) is P-provable. 2

Godels Completeness Theorem In any first-order predicate calculus, the

theorems are precisely the logically valid well-formed formulas (i. e. those thatare true in every model of the calculus). 2

Since [(A A) A] is a theorem of first-order logic70, we have, byModus Ponens, that:

Lemma 25 ((x)R(x)) is P-provable. 2

Now, Godel also showed that:

Lemma 26 If P is consistent, then [R(n) ] is P-provable for any given P-numeral [n]71. 2

It follows that:

Theorem 6 If P is consistent, then it is not-consistent. 2

Since Godels argument holds in PA, we further have that:

Theorem 7 If PA is consistent, then it is not-consistent.72 2

11 Appendix G: Churchs Thesis as a weaker

equivalence

The reason Churchs thesis does not hold is that the thesis is expressed as a

strong, refutable, identity rather than the weakerand intuitively more plausi-ble equivalence:

Thesis 1 (Weak Churchs Thesis) A number-theoretic function (or relation,treated as a Boolean function) is effectively computable if, and only if, it isinstantiationally equivalent to a partial-recursive function. 2

70See [Me64], p.32, Ex.1.71[Go31], p.26(2).72See also [An08].

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It is significant that Godel (initially) and Church (subsequentlypossibly

under the influence of Godels disquietitude) enunciated Churchs formulation ofeffective computability as a Thesis because Godel was instinctively uncomfort-able with accepting it as a definition that fully captures the essence of intuitiveeffective computability73.

Godels reservations seem vindicated if we accept that a number-theoreticfunction can be effectively computable instantiationally, but not algorithmically.

The possibility that truth may be be effectively computable instantiation-ally, but not algorithmically, is implicit in Godels famous 1951 Gibbs lecture74,where he remarks:

I wish to point out that one may conjecture the truth of a univer-sal proposition (for example, that I shall be able to verify a certainproperty for any integer given to me) and at the same time conjec-

ture that no general proof for this fact exists. It is easy to imag-ine situations in which both these conjectures would be very wellfounded. For the first half of it, this would, for example, be the caseif the proposition in question were some equation F(n) = G(n) oftwo number-theoretical functions which could be verified up to verygreat numbers n.75

Such a possibility is also implicit in Turings remarks76:

The computable numbers do not include all (in the ordinary sense)definable numbers. Let P be a sequence whose n-th figure is 1 or0 according as n is or is not satisfactory. It is an immediate conse-quence of the theorem of 8 that P is not computable. It is (so far as

we know at present) possible that any assigned number of figures ofP can be calculated, but not by a uniform process. When sufficientlymany figures of P have been calculated, an essentially new methodis necessary in order to obtain more figures.

The need for placing such a distinction on a formal basis has also beenexpressed explicitly on occasion77. Thus, Boolos, Burgess and Jeffrey78 define adiagonal function, d, any value of which can be computed effectively, althoughthere is no single algorithm that can effectively compute d.

Now, the straightforward way of expressing this phenomenon should be tosay that there are well-defined number-theoretic functions that are instantiation-ally computable, but not algorithmically computable79. Yet, following Churchand Turing, such functions are labeled as effectively uncomputable80!

73See [Si97].74[Go51].75Parikhs paper [Pa71] can also be viewed as an attempt to investigate the consequences

of expressing the essence of Godels remarks formally.76[Tu36], 9, para II.77Parikhs distinction between decidability and feasibility in [Pa71] also appears to echo

the need for such a distinction.78[BBJ03], p. 37.79Or, preferably, one could borrow the analogous terminology from the theory of functions

of real and complex variables and term such functions as computable, but not uniformlycomputable.

80The issue here seems to be that, when using language to express the abstract objects ofour individual, and common, mental concept spaces, we use the word exists loosely in threesenses, without making explicit distinctions between them (see [An07]).

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According to Turings Thesis, since d is not Turing-computable, d

cannot be effectively computable. Why not? After all, although noTuring machine computes the function d, we were able to compute atleast its first few values, For since, as we have noted, f1 = f1 = f1 =the empty function we have d(1) = d(2) = d(3) = 1. And it mayseem that we can actually compute d(n) for any positive integernif we dont run out of time.81

The reluctance to treat a function such as d(n)or the function (n) thatcomputes the nth digit in the decimal expression of a Chaitin constant 82ascomputable, on the grounds that the time needed to compute it increases mono-tonically with n, is curious83; the same applies to any total Turing-computablefunction f(n)84!

References

[Be59] Evert W. Beth. 1959. The Foundations of Mathematics. Studies in Logic and theFoundations 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 andLogic (4th ed). Cambridge University Press, Cambridge.

[Br08] L. E. J. Brouwer. 1908. The Unreliability of the Logical Principles. English trans-lation in A. Heyting, Ed. L. E. J. Brouwer: Collected Works 1: Philosophy andFoundations of Mathematics. Amsterdam: North Holland / New York: AmericanElsevier (1975): pp. 107-111.

[Br13] L. E. J. Brouwer. 1913. Intuitionism and Formalism. Inaugural address at the Uni-

versity of Amsterdam, October 14, 1912. Translated by Professor Arnold Dresdenfor the Bulletin of the American Mathematical Society, Volume 20 (1913), pp.81-96.1999. Electronically published in Bulletin (New Series) of the American Mathemat-ical Society, Volume 37, Number 1, pp.55-64.

[Br23] L. E. J. Brouwer. 1923. On the significance of the principle of the excluded middle inmathematics, especially in function theory. Address delivered on 21 September 1923at the annual convention of the Deutsche Mathematiker-Vereinigung in Marburgan der Lahn. In Jean van Heijenoort. 1967. Ed. From Frege to Godel: A sourcebook in Mathematical Logic, 1878 - 1931. Harvard University Press, Cambridge,Massachusetts.

[Br27] L. E. J. Brouwer. 1927. Intuitionistic reflections on formalism. In Jean van Hei-jenoort. 1967. Ed. From Frege to Godel: A source book in Mathematical Logic, 1878- 1931. Harvard University Press, Cambridge, Massachusetts.

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

81[BBJ03], p.37.82Chaitins Halting Probability is given by 0 < =

2|p| < 1, where the summation is

over all self-delimiting programs p that halt, and |p| is the size in bits of the halting programp; see [Ct75].

83The incongruity of this is addressed by Parikh in [Pa71].84The 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, weknow we may need distinctly different programs for computing f(n) for different values ofn,where the length of the program will, sometime, reference n.

19


20/21

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

[Cook] Stephen Cook. The P versus NP Problem. Official description provided for the ClayMathematical Institute, Cambridge, Massachusetts.

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

[EC89] Richard L. Epstein, Walter A. Carnielli. 1989. Computability: Computable Func-tions, Logic, and the Foundations of Mathematics. Wadsworth & Brooks, Califor-nia.

[Fo09] Lance Fortnow. 2009. The Status of the P Versus NP Problem. Communications ofthe ACM, Volume 52, Issue 9 (September 2009).

[Go31] Kurt Godel. 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 Godel. 1951. Some basic theorems on the foundations of mathematics andtheir implications. Gibbs lecture. In Kurt Godel, Collected Works III, pp.304-323.1995. Unpublished Essays and Lectures. Solomon Feferman et al (ed.). Oxford Uni-versity Press, New York.

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

[HA28] David Hilbert & Wilhelm Ackermann. 1928. Principles of Mathematical Logic.Translation of the second edition of the Grundzuge 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 Munster on4th June 1925 at a meeting of the Westphalian Mathematical Society. In Jean vanHeijenoort. 1967. Ed. From Frege to Godel: A source book in Mathematical Logic,1878 - 1931. Harvard University Press, Cambridge, Massachusetts.

[Hi27] David Hilbert. 1927. The Foundations of Mathematics. Text of an address deliveredin July 1927 at the Hamburg Mathematical Seminar. In Jean van Heijenoort. 1967.Ed. From Frege to Godel: A source book in Mathematical Logic, 1878 - 1931.Harvard University Press, Cambridge, Massachusetts.

[Ho01] Wilfrid Hodges. 2001. Tarskis Truth Definitions. Stanford Encyclopedia ofPhilosophy. (Winter 2001 Edition). Edward N. Zalta (ed.). (Web essay:http://plato.stanford.edu/entries/tarski-truth/.

[Kl52] Stephen Cole Kleene. 1952. Introduction to Metamathematics. North Holland Pub-lishing Company, Amsterdam.

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

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

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

[Ra02] Ran Raz. 2002. P=NP, Propositional Proof Complexity, and Resolution LowerBounds for the Weak Pigeonhole Principle. ICM 2002, Vol. III, 1-3.

[Rg87] Hartley Rogers Jr. 1987. Theory of Recursive Functions and Effective Computabil-ity. MIT Press, Cambridge, Massachusetts.

20
http://plato.stanford.edu/entries/tarski-truth/http://plato.stanford.edu/entries/tarski-truth/


21/21

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

[Si97] Wilfried Sieg. 1997. Step by recursive step: Churchs analysis of effective calcula-bility. Bulletin of Symbolic Logic, Volume 3, Number 2.

[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.

[Tu36] Alan Turing. 1936. On computable numbers, with an application to the Entschei-dungsproblem. 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 Com-pany, Amsterdam.

[We27] Hermann Weyl. 1927. Comments on Hilberts second lecture on the foundations ofmathematics. In Jean van Heijenoort. 1967. Ed. From Frege to Godel: A sourcebook in Mathematical Logic, 1878 - 1931. Harvard University Press, Cambridge,Massachusetts.

[An07] Bhupinder Singh Anand. 2007. Why we shouldnt fault Lucas and Penrose for con-tinuing to believe in the Godelian argument against computationalism - II. TheReasoner, Vol(1)7 p2-3.

[An08] . . . . 2008. Why Brouwer was justified in his objection to Hilberts unqualified in-terpretation of quantification. Proceedings of the 2008 International Conference onFoundations of Computer Science, July 14-17, 2008, Las Vegas, USA.

[An09a] .. .. 2009. The significance of Aristotles particularisation in the foundations ofmathematics, logic and computability II: Godel and formally undecidable arith-

metical propositions. Proceedings of the 2009 International Conference on Theoret-ical and Mathematical Foundations of Computer Science, July 13-16, Orlando, FL,USA.

[An09b] .. .. 2009. The significance of Aristotles particularisation in the foundations ofmathematics, logic and computability III: Rosser and formally undecidable arith-metical propositions. Proceedings of the 2009 International Conference on Founda-tions of Computer Science, July 13-16, Las Vegas, NV, USA.

[An09c] .. .. 2009. The significance of Aristotles particularisation in the foundations ofmathematics, logic and computability IV: Turing and a sound, finitary, interpre-tation of PA. Proceedings of the 2009 International Conference on Theoretical andMathematical Foundations of Computer Science, July 13-16 2009, Orlando, FL,USA.

Authors postal address: 32 Agarwal House, D Road, Churchgate, Mumbai - 400 020, Maharashtra, India. Email:

[email protected], [email protected].

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