30 Finitary Logic PA Update

8/2/2019 30 Finitary Logic PA Update

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Some consequences of interpreting the associated

logic of the first-order Peano Arithmetic PA

finitarily

Bhupinder Singh Anand

Draft of January 17, 2012. An earlier version of this manuscript is arXived here.

Abstract

We show that Tarskis inductive definitions actually allow us to de-fine the satisfaction and truth of the quantified formulas of the first-orderPeano Arithmetic PA constructively over the domain N of the naturalnumbers in two essentially different ways: (a) in terms of algorithmic ver-ifiabilty; and (b) in terms of algorithmic computability. We show thatthe standard interpretation of PA defines the satisfaction and truth of theformulas of the first-order Peano Arithmetic PA constructively in terms ofalgorithmic verifiability. It is accepted that this interpretation cannot layclaim to be finitary; it does not lead to a finitary justification of the AxiomSchema of (finite) Induction of PA from which we may concludein an in-tuitionistically unobjectionable mannerthat PA is consistent. However,we now show that the PA-axiomsincluding the Axiom Schema of (finite)Inductionare algorithmically computable as satisfied / true under thestandard interpretation of PA, and that the PA rules of inference preservealgorithmically computable satisfiability / truth under the standard inter-pretation. We conclude that the algorithmically computable PA-formulascan provide a finitary interpretation of PA from which we may concludethat PA is consistent in an intuitionistically unobjectionable manner. Wedefine such an interpretation, and show that if the associated logic is in-terpreted finitarily then (i) PA is categorical and (ii) Godels Theorem VIholds vacuously in PA since PA is consistent but not -consistent. Thisreflects the fact that PA is -consistent if, and only if, Aristotles par-ticularisation is presumed to always hold under any interpretation of theassociated logic; and that the standard interpretation of PA is a model of

PA if, and only if, PA is -consistent.Keywords Algorithmic computability, algorithmic verifiability, Aristotles particu-larisation, consistency, first-order, -consistency, Peano Arithmetic PA, satisfaction,soundness, standard interpretation, Tarski.

1 Introduction

We formally define what it means for a formula of an arithmetical language tobe:

Subject class: LO; MSC: 03B10

1
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(i) Algorithmically verifiable (Definition 12);

(ii) Algorithmically computable (Definition 13).

under an interpretation.

We argue separately1 that Godels arithmetical representation of any recursiverelation yields a class of PA formulas that are algorithmically verifiable but notalgorithmically computable in the above sense.

We then show that:

(a) The PA-formulas are decidable under the standard interpretationof PA if, and only if, they are algorithmically verifiable under theinterpretation (Corollary 2);

Although the standard interpretation is believed to define a modelof PA, the definition cannot claim to be finitary since it does notlead to a finitary justification of the Axiom Schema of (finite) In-duction of PA from which we may concludein an intuitionisticallyunobjectionable mannerthat PA is consistent2. We note furtherthat Gerhard Gentzens constructive3 consistency proof for formalnumber theory4 is debatably finitary5, since it involves a Rule ofInfinite Induction that admits appeal to the properties of transfiniteordinals; and we show in Section 12, Appendix E that we cannot in-troduce a transfinite ordinal into any model of PA without invitinginconsistency.

(b) The PA-axioms are algorithmically computable as satisfied / trueunder the standard interpretation of PA (Lemmas 4 and 5);

(c) Generalisation and Modus Ponens preserve algorithmically com-putable truth under the standard interpretation of PA (Lemmas 6and 7);

(d) The provable PA-formulas are precisely the ones that are algo-rithmically computable as satisfied / true under the standard inter-pretation of PA (Theorem 4).

We conclude that the algorithmically computable PA-formulas can providea soundin the sense of Definition 10finitary interpretation of PA (Theorem5).

We note that PA is -consistent if, and only if, Aristotles particularisa-tion (Definition 1) is presumed to always hold under any interpretation of theassociated logic (Section 8, Appendix A).

We then show that if classical first-order logic is interpreted finitarily (Sec-tion 10, Appendix C) without the presumption that Aristotles particularisationnecessarily holds under the interpretation, then we may conclude that:

1See http://alixcomsi.com/32 Undecidable Arithmetical Propositions Update.pdf.2The possibility/impossibility of such justification was the sub ject of the famous Poincare-

Hilbert debate. See [Hi27], p.472; also [Br13], p.59; [We27], p.482; [Pa71], p.502-503.3In the sense highlighted by Elliott Mendelson in [Me64], p.261.4cf. [Me64], p258.5See for instance http://en.wikipedia.org/wiki/Hilberts program.

2
http://alixcomsi.com/32_Undecidable_Arithmetical_Propositions_Update.pdfhttp://alixcomsi.com/32_Undecidable_Arithmetical_Propositions_Update.pdfhttp://en.wikipedia.org/wiki/Hilbert's_programhttp://en.wikipedia.org/wiki/Hilbert's_programhttp://alixcomsi.com/32_Undecidable_Arithmetical_Propositions_Update.pdf


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(f) PA is consistent (Theorem 6);

(g) PA is categorical (Corollary 3);

(h) PA is not -consistent (Corollary 5);

(i) the standard interpretation of PA is not sound6, and does notyield a model for PA (Corollary 6).

1.1 Comments, Notation and Definitions

Comments We have taken some liberty in emphasising standard definitionsselectively, and interspersing our arguments liberally with comments and ref-erences, generally of a foundational nature. These are intended to reflect ourunderlying thesis that essentially arithmetical problems appear more natural

when expressedand viewedwithin the perspective of an interpretation ofPA that appeals to the evidence provided by a deterministic Turing machine7

along the lines suggested in Section 4; a perspective that, by its very nature,cannot appeal implicitly to transfinite concepts.

Evidence It is by now folklore . . . that one can view the values of a simplefunctional language as specifying evidence for propositions in a constructivelogic . .. 8.

Notation We use square brackets to indicate that the contents represent asymbol or a formulaof a formal theorygenerally assumed to be well-formedunless otherwise indicated by the context.9 We use an asterisk to indicate thatthe associated expression is to be interpreted semantically with respect to some

well-defined interpretation.

Definition 1 Aristotles particularisation This holds that from a meta-assertion such as:

It is not the case that: For any given x, P(x) does not hold,

usually denoted symbolically by (x)P(x), we may always validly infer inthe classical, Aristotlean, logic of predicates10 that:

There exists an unspecified x such that P(x) holds,

usually denoted symbolically by (x)P(x).

The significance of Aristotles particularisation for the first-order pred-

icate calculus: We note that in a formal language the formula [(x)P(x)] is anabbreviation for the formula [(x)P(x)]. The commonly accepted interpre-tation of this formulaand a fundamental tenet of classical logic unrestrictedly

6In the sense of Definition 10.7A deterministic Turing machine has only one possible move from a given configuration.8[Mu91].9In 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 accordancewith specific rules for such formation and manipulationin the manner of a mechanical orelectronic devicewithout any regards to what the symbolism might represent semanticallyunder an interpretation that gives them meaning.

10[HA28], pp.58-59.

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adopted as intuitively obvious by standard literature11 that seeks to build uponthe formal first-order predicate calculustacitly appeals to Aristotlean particu-larisation.

However, L. E. J. Brouwer had noted in his seminal 1908 paper on the unreli-ability of logical principles12 that the commonly accepted interpretation of thisformula is ambiguous if interpretation is intended over an infinite domain.

Brouwer essentially argued that, even supposing the formula [P(x)] of a for-mal Arithmetical language interprets as an arithmetical relation denoted byP(x), and the formula [(x)P(x)] as the arithmetical proposition denotedby (x)P(x), the formula [(x)P(x)] need not interpret as the arithmeti-cal proposition denoted by the usual abbreviation (x)P(x); and that suchpostulation is invalid as a general logical principle in the absence of a means forconstructing some putative object a for which the proposition P(a) holds inthe domain of the interpretation.

Hence we shall follow the convention that the assumption that (x)P(x)is the intended interpretation of the formula [(x)P(x)]which is essentially

the assumption that Aristotles particularisation holds over the domain of theinterpretationmust always be explicit.

The significance of Aristotles particularisation for PA: In order to avoidintuitionistic objections to his reasoning, Kurt Godel introduced the syntacticproperty of -consistency13 as an explicit assumption in his formal reasoning inhis seminal 1931 paper on formally undecidable arithmetical propositions14.

Godel explained at some length15 that his reasons for introducing -consistencyexplicitly was to avoid appealing to the semantic concept of classical arithmeticaltruth in Aristotles logic of predicates (which presumes Aristotles particularisa-tion).

We show in Section 8, Appendix A that the two concepts are meta-mathematicallyequivalent in the sense that, if PA is consistent, then PA is -consistent if, andonly if, Aristotles particularisation holds under the standard interpretation ofPA.

Definition 2 The structure N The structure of the natural numbersnamely,{N (the set of natural numbers); = (equality); (the successor function); + (theaddition function); (the product function); 0 (the null element)}.

Definition 3 The axioms of first-order Peano Arithmetic (PA)

PA1 [(x1 = x2) ((x1 = x3) (x2 = x3))];PA2 [(x1 = x2) (x

1 = x

2)];

PA3 [0 = x1];

PA4 [(x1 = x

2) (x1 = x2)];

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

2) = (x1 + x2)

];PA

7[(x

1 0) = 0];

PA8 [(x1 x2) = ((x1 x2) + x1)];

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

11See [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.

12[Br08].13The significance of -consistency for the formal system PA is highlighted in Section 8,

Appendix A.14[Go31], p.23 and p.28.15In his introduction on p.9 of [Go31].

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Definition 4 Generalisation in PA If[A] is PA-provable, then so is [(x)A].

Definition 5 Modus Ponens in PA If [A] and [A B] are PA-provable,then so is [B].

Definition 6 Standard interpretation of PA The standard interpretationIP A(N, S tandard) of PA over the structure N is the one in which the logical con-stants have their usual interpretations16 in Aristotles logic of predicates (whichsubsumes Aristotles particularisation), and17:

(a) the set of non-negative integers is the domain;(b) the symbol [0] interprets as the integer 0;(c) the symbol [] interprets as the successor operation (addition of 1);(d) the symbols [+] and [] interpret as ordinary addition and multiplication;

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

Definition 7 Simple consistency: A formal system S is simply consistentif, and only if, there is no S-formula [F(x)] for which both [(x)F(x)] and[(x)F(x)] are S-provable.

Definition 8 -consistency: A formal system S is -consistent if, and onlyif, there is no S-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].

Definition 9 Soundness (formal system - non-standard): A formal sys-tem S is sound under an interpretation IS with respect to a domainD if, andonly if, every theorem [T] of S translates as [T] is true underIS inD.

Definition 10 Soundness (interpretation - non-standard): An interpre-tationIS of a formal system S is sound with respect to a domainD if, and onlyif, S is sound under the interpretation IS over the domainD.

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 defined asa model18 of the formal system. This definition suffers, however, from an implicitcircularity: the formal logic L underlying any interpretation of S is implicitly assumedto be sound. The above definitions seek to avoid this implicit circularity by delinkingthe defined soundness of a formal system under an interpretation from the implicitsoundness of the formal logic underlying the interpretation. This admits the casewhere, even if L1 and L2 are implicitly assumed to be sound, S + L1 is sound, butS + L2 is not. Moreover, an interpretation of S is now a model for S if, and only if,it is sound.19

Definition 11 Categoricity: A formal system S is categorical if, and onlyif, it has a sound20 interpretation and any two sound interpretations of S areisomorphic.21

16These are expressed formally in Section 9, Appendix B, and essentially follow the defini-tions in [Me64], p.49.

17See [Me64], p.107.18We follow the definition in [Me64], p.51.19My thanks to Professor Rohit Parikh for highlighting the need for making such a distinc-

tion explicit.20In the sense of Definitions 9 and 10.21Compare [Me64], p.91.

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2 Interpretation of an arithmetical language in

terms of Turing computabilityWe begin by noting that we can, in principle, define22 the classical satisfactionand truth of the formulas of a first order arithmetical language, such as PA,verifiably under an interpretation using as evidence 23 the computations of asimple functional language.

Such definitions follow straightforwardly for the atomic formulas of the lan-guage (i.e., those without the logical constants that correspond to negation,conjunction, implication and quantification) from the standard definition ofa deterministic Turing machine24, based essentially on Alan Turings seminal1936 paper on computable numbers25.

Moreover, it follows from Alfred Tarskis seminal 1933 paper on the the con-

cept of truth in the languages of the deductive sciences

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that the satisfactionand truth of those formulas of a first-order language which contain logicalconstants can be inductively defined, under an interpretation, in terms of thesatisfaction and truth of the interpretations of only the atomic formulas ofthe language.

Hence the satisfaction and truth of those formulas (of an arithmetical lan-guage) which contain logical constants can, in principle, also be defined verifiablyunder an interpretation using as evidence the computations of a deterministicTuring machine.

We show in Section 4 that this is indeed the case for PA under the standardinterpretation IP A(N, S tandard), when this is explicitly defined as in Section 5.

We show, moreover, that we can further define algorithmic truth and al-gorithmic falsehood under IP A(N, S tandard) such that the PA axioms interpret

as always algorithmically true, and the rules of inference preserve algorithmictruth.

Significance of algorithmic truth: The algorithmically true propositionsofN under IP A(N, S tandard) are, moreover, a proper subset

27 of the verifiablytrue propositions ofN under IP A(N, Standard); they suggest a possible finitarymodel of PA that establishes the consistency of PA constructively.

2.1 The definitions of algorithmic truth and algorithmicfalsehood under IP A(N, Standard) are not symmetric withrespect to truth and falsehood under IP A(N, Standard)

However, the definitions of algorithmic truth and algorithmic falsehood under

IP A(N, Standard) are not symmetric with respect to classical (verifiable) truthand falsehood under IP A(N, S tandard).

For instance, if a formula [(x)F(x)] of an arithmetic is algorithmically trueunder an interpretation (such as IP A(N, S tandard)), then we may conclude thatthere is a deterministic Turing machine that computes28 [F(x)] and provides

22Formal definitions are given in Section 4.23[Mu91].24[Me64], pp.229-231.25[Tu36].26[Ta33].27This follows immediately from Corollary 4.28In the non-standard sense of Section 3 below.

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evidence that, for any given numeral [a], the formula [F(a)] is algorithmically

true under the interpretation.In other words, there is a deterministic Turing machine that computes[F(x)] and can provide evidence that the interpretation F(a) of [F(a)] holdsin N for any given natural number a.

Defining the term hold: We define the term holdwhen used in con-nection with an interpretation of a formal language and, more specifically, withreference to the operations of a deterministic Turing machine associated with theatomic formulas of the languageexplicitly in Section 4; the aim being to avoidappealing to the classically subjective (and existential) connotation implicitlyassociated with the term under an implicitly defined standard interpretation ofan arithmetic29.

However, if a formula [(x)F(x)] of an arithmetic is algorithmically falseunder an interpretation, then we can only conclude that there is no deterministicTuring machine that can compute [F(x)] and provide evidence whether theinterpretation F(a) holds or not in N for any given natural number a.

We cannot conclude (since [F(x)] may be a Halting type of relation) thatthere is a numeral [a] such that the formula [F(a)] is algorithmically false underthe interpretation; nor can we conclude that there is a natural number b suchthat F(b) does not hold in N.

Such a conclusion would require:

(i) either some additional evidence that will verify for some assignment of nu-merical values to the free variables of [F] that the corresponding interpretationF does not hold30;

(ii) or the additional assumption that either Aristotles particularisation31

holds over the domain of the interpretation (as is implicitly presumed under thestandard interpretation of PA) or that the arithmetic is -consistent32.

3 Defining algorithmic verifiability and algorith-mic computability

The asymmetry of Section 2.1 suggests the following two concepts33:

Definition 12 Algorithmic verifiability:

An arithmetical formula [(x)F(x)] is algorithmically verifiable as true underan interpretation if, and only if, for any given numeral [a], we can define a

deterministic Turing machine TM that computes [F

(a

)] in the Tarskian sense(as defined in Section 4) such that TM will halt on null input if, and only if,[F(a)] interprets as true under the interpretation.

Tarskian interpretation of an arithmetical language verifiably in terms ofTuring computability We show in Section 4 that the algorithmic verifiability of

29As, for instance, in [Go31].30Essentially reflecting Brouwers objection to the assumption of Aristotles particularisation

over an infinite domain.31Definition 1.32An assumption explicitly introduced by Godel in [Go31].33My thanks to Dr. Chaitanya H. Mehta for advising that the focus of this investigation

should be the distinction between these two concepts.

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the formulas of a formal language which contain logical constants can be inductivelydefined under an interpretation in terms of the algorithmic verifiability of the inter-

pretations of the atomic formulas of the language; further, that the PA-formulas aredecidable under the standard interpretation of PA if, and only if, they are algorith-mically verifiable under the interpretation (Corollary 2).

Definition 13 Algorithmic computability:

An arithmetical formula [(x)F(x)] is algorithmically computable as true underan interpretation if, and only if, we can define a deterministic Turing machineTM that computes [F(x)] in the Tarskian sense (as defined in Section 4) suchthat, for any given numeral [a], TM will halt on [a] if, and only if, [F(a)]interprets as true under the interpretation.

Tarskian interpretation of an arithmetical language algorithmically in termsof Turing computability We show in Section 4 that the algorithmic computabilityof the formulas of a formal language which contain logical constants can also beinductively defined under an interpretation in terms of the algorithmic computabilityof the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under an algorithmic interpretation of PA if, and only if, theyare algorithmically computable under the interpretation .

We now show that the above concepts are well-defined under the standardinterpretation of PA.

4 The implicit Satisfaction condition in Tarskisinductive assignment of truth-values under aninterpretation

We first consider the significance of the implicit Satisfaction condition in Tarskisinductive assignment of truth-values under an interpretation.We note thatessentially following standard expositions34 of Tarskis in-

ductive definitions on the satisfiability and truth of the formulas of a formallanguage under an interpretationwe can define:

Definition 14 If [A] is an atomic formula [A(x1, x2, . . . , xn)] of a formal lan-guage S, then the denumerable sequence (a1, a2, . . .) in the domain D of aninterpretationIS(D) of S satisfies [A] if, and only if:

(i) [A(x1, x2, . . . , xn)] interprets under IS(D) as a unique relationA(x1, x2, . . . , xn) inD for any witness WD ofD;

(ii) there is a Satisfaction Method, SM(IS(D

)) that provides objective

evidence35 by which any witness WD ofD can objectively define forany atomic formula [A(x1, x2, . . . , xn)] of S, and any given denumer-able sequence (b1, b2, . . .) ofD, whether the propositionA

(b1, b2, . . . , bn)holds or not inD;

(iii) A(a1, a2, . . . , an) holds inD for anyWD.

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

34cf. [Me64], p.51.35In the sense of [Mu91].

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

A constructive perspective: We highlight the word define in (ii) above toemphasise the constructive perspective underlying this paper; which is that theconcepts of satisfaction and truth under an interpretation are to be explicitlyviewed as objective assignments by a convention that is witness-independent.A Platonist perspective would substitute decide for define, thus implicitlysuggesting that these concepts can exist, in the sense of needing to be discoveredby some witness-dependent meanseerily akin to a revelationif the domainD is N.

We can now inductively assign truth values of satisfaction, truth, and fal-sity to the compound formulas of a first-order theory S under the interpretationIS(D) in terms of only the satisfiability of the atomic formulas of S over D as

usual

36

:

Definition 15 A denumerable sequence s of D satisfies [A] under IS(D) if,and only if, s does not satisfy [A];

Definition 16 A denumerable sequence s ofD satisfies [A B] under IS(D)if, and only if, either it is not the case that s satisfies [A], or s satisfies [B];

Definition 17 A denumerable sequence s of D satisfies [(xi)A] under IS(D)if, and only if, given any denumerable sequence t ofD which differs from s inat most the ith component, t satisfies [A];

Definition 18 A well-formed formula [A] ofD is true underIS(D) if, and onlyif, given any denumerable sequence t ofD, t satisfies [A];

Definition 19 A well-formed formula [A] ofD is false underIS(D) if, and onlyif, it is not the case that [A] is true underIS(D).

It follows that37:

Theorem 1 (Satisfaction Theorem) If, for any interpretation IS(D) of a first-order theory S, there is a Satisfaction Method SM(IS(D)) which holds for awitness WD ofD, then:

(i) The 0 formulas of S are decidable as either true or false overD underIS(D);

(ii) If the n formulas of S are decidable as either true or as falseoverD underIS(D), then so are the (n + 1) formulas of S.

Proof It follows from the above definitions that:

(a) If, for any given atomic formula [A(x1, x2, . . . , xn)] of S, it is decidableby WD whether or not a given denumerable sequence (a1, a2, . . .) of D satis-fies [A(x1, x2, . . . , xn)] in D under IS(D) then, for any given compound formula[A1(x1, x2, . . . , xn)] of S containing any one of the logical constants ,,, it

36See [Me64], p.51; [Mu91].37cf. [Me64], pp.51-53.

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is decidable by WD whether or not (a1, a2, . . .) satisfies [A1(x1, x2, . . . , xn)] in D

under IS(D);(b) If, for any given compound formula [Bn(x1, x2, . . . , xn)] of S containing

n of the logical constants ,,, it is decidable by WD whether or not a givendenumerable sequence (a1, a2, . . .) ofD satisfies [B

n(x1, x2, . . . , xn)] in D underIS(D) then, for any given compound formula [B

(n+1)(x1, x2, . . . , xn)] of S con-taining n + 1 of the logical constants ,, , it is decidable by WD whether ornot (a1, a2, . . .) satisfies [B

(n+1)(x1, x2, . . . , xn)] in D under IS(D);

We thus have that:

(c) The 0 formulas of S are decidable by WD as either true or false over Dunder IS(D);

(d) If the n formulas of S are decidable by WD as either true or as false

overD

under IS(D), then so are the (n + 1) formulas of S.2

In other words, if the atomic formulas of of S interpret under IS(D) as decid-able with respect to the Satisfaction Method SM(IS(D)) by a witness WD oversome domain D, then the propositions of S (i.e., the n and n formulas of S)also interpret as decidable with respect to SM(IS(D)) by the witness WD overD.

We now consider the application of Tarskis definitions to various interpre-tations of first-order Peano Arithmetic PA.

4.1 The standard interpretation of PA

The standard interpretation IP A(N, Standard) of PA is obtained if, in IS(D):

(a) we define S as PA with standard first-order predicate calculus asthe underlying logic38;

(b) we define D as N;

(c) for any atomic formula [A(x1, x2, . . . , xn)] of PA and sequence(a1, a2, . . . , an) ofN, we take SATCON(IP A(N)) as:

A(a1, a2, . . . , a

n) holds in N and, for any given sequence

(b1, b2, . . . , b

n) of N, the proposition A

(b1, b2, . . . , b

n) is

decidable in N;

(d) we define the witness W(N, S tandard) informally as the math-ematical intuition of a human intelligence for whom, classically,

SATCON(IP A(N)) has been implicitly accepted as objectively de-cidable in N;

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

Lemma 1 A(x1, x2, . . . , xn) is both algorithmically verifiable andalgorithmically computable inN byW(N, S tandard).

38Where the string [( . . .)] is defined asand is to be treated as an abbreviation forthe string [( . . .)]. We do not consider the case where the underlying logic is Hilbertsformalisation of Aristotles logic of predicates in terms of his -operator ([Hi27], pp.465-466).

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Proof (i) It follows from the argument in Theorem 2 (below) thatA(x1, x2, . . . , xn) is algorithmically verifiable in N byW(N, S tandard).

(ii) It follows from the argument in Theorem 3 (below) that A(x1, x2,. . . , xn) is algorithmically computable in N by W(N, Standard). Thelemma follows. 2

Now, although it is not immediately obvious from the standard in-terpretation of PA which of (i) or (ii) may be taken for explicitly de-ciding SATCON(IP A(N)) by the witness W(N, Standard), we shallshow in Section 4.3 that (i) is consistent with (e) below; and inSection 4.5 that (ii) is inconsistent with (e). Thus the standardinterpretation of PA implicitly presumes (i).

(e) we postulate that Aristotles particularisation holds over N39.

Clearly, (e) does not form any part of Tarskis inductive definitions of the

satisfaction, and truth, of the formulas of PA under the above interpretation.Moreover, its inclusion makes IP A(N, Standard) extraneously non-finitary

40.

The question arises: Can we formulate the standard interpretation of PA with-out assuming (e) extraneously?

We answer this question affirmatively in Section 4.3 where:

(1) We replace the mathematical intuition of a human intelligence by definingan objective witness W(N, Instantiational) as the meta-theory MP A of PA;

(2) We show that W(N, Instantiational) can decide whether c = d is true or

false by instantiationally computing the Boolean function A(x1, x2, . . . , xn) forany given sequence of natural numbers (b1, b

2, . . . , b

n);

(3) We show that this yields an instantiational interpretation of PA over [N] thatis sound if, and only if, (e) holds.

(4) W(N, Instantiational) is thus an instantiational formulation of the standardinterpretation of PA over N (which is presumed to be sound).

We note further that if PA is -inconsistent, then Aristotles particularisa-tion does not hold over N, and the interpretation IP A(N, S tandard) is not sound.

4.2 Godels non-standard interpretation of PA

A non-standard (Godelian) interpretation IP A(N, Nonstandard) of a putative-consistent PA is obtained if, in IS(D):

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

(b) we define D as an undefined extension N ofN;

(c) for any atomic formula [A(x1, x2, . . . , xn)] of PA and sequence(a1, a2, . . . , an) ofN, we take SATCON(IP A(N)) as:

A(a1, a2, . . . , a

n) holds in N and, for any given se-

quence (b1, b2, . . . , b

n) ofN, the proposition A

(b1, b2, . . . , b

n)

is decidable as either holding or not holding in N;

39Hence a PA formula such as [(x)F(x)] interprets under IP A(N, Standard) as There issome natural number n such that F(n) holds in N.

40[Br08].

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(d) we postulate that SATCON(IP A(N)) is always decidable by a

putative witness WN

, and that WN

can, further, determine somenumbers in N which are not natural numbers;

(e) we assume that PA is -consistent.

Clearly, the interpretation IP A(N, Nonstandard) of a putative -consistentPA cannot claim to be finitary. Moreover, if PA is -inconsistent, then theGodelian non-standard interpretation IP A(N, Nonstandard) of PA is also notsound41.

4.3 An instantiational interpretation of PA in PA

We next consider the instantiational interpretation42 IP A(N, Instantiational) ofPA where:


(b) we define D as PA;

(c) for any atomic formula [A(x1, x2, . . . , xn)] of PA and any se-quence [(a1, a2, . . . , an)] of PA numerals, we take SATCON(IP A(PA))as:

[A(a1, a2, . . . , an)] is provable in PA and, for any givensequence of numerals [(b1, b2, . . . , bn)] of PA, the formula[A(b1, b2, . . . , bn)] is decidable as either provable or notprovable in PA;

(d) we define the witness W(N, Instantiational) as the meta-theoryMP A of PA.

Lemma 2 [A(x1, x2, . . . , xn)] is always algorithmically verifiable inPA byW(N, Instantiational).

Proof It follows from Godels definition of the primitive recur-sive relation xBy43where x is the Godel number of a proof se-quence in PA whose last term is the PA formula with G odel-numberythat, if [A(x1, x2, . . . , xn)] is an atomic formula of PA, MP Acan algorithmically verify for any given sequence [(b1, b2, . . . , bn)] ofPA numerals which one of the PA formulas [A(b1, b2, . . . , bn)] and[A(b1, b2, . . . , bn)] is necessarily PA-provable. 2

Now, if PA is consistent but not -consistent, then there is a Godelian for-mula [R(x)] such that (see Section 7):

(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.

41In which case we cannot validly conclude from Godels formal reasoning in ([Go31]) thatPA must have a non-standard model.

42The raison detre, and significance, of such interpretation is outlined in this short note.43[Go31], p. 22(45).

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However, if IP A(N, I nstantiational) is sound, then (ii) implies contradictorily

that it is not the case that, for any given numeral [n], [R(n)] is PA-provable.It follows that ifIP A(N, I nstantiational) is sound then PA is -consistent and,

ipso facto, Aristotles particularisation must hold over N.Moreover, if PA is consistent, then every PA-provable formula interprets as

true under some sound interpretation of PA. Hence MP A can effectively decidewhether, for any given sequence of natural numbers (b1, b

2, . . . , b

n) in N, the

proposition A(b1, b2, . . . , b

n) holds or not in N.

It follows that IP A(N, Instantiational) is an instantiational formulation of thestandard interpretation of PA in which we do not need to extraneously assumethat Aristotles particularisation holds over N.

The interpretation IP A(N, I nstantiational) is of interest because, if it were asound interpretation of PA, then PA would establish its own consistency44!

4.4 A set-theoretic interpretation of PA

We consider next a set-theoretic interpretation IP A(ZF, Cantor) of PA over thedomain of ZF sets, which is obtained if, in IS(D):


(b) we define D as ZF;

(c) for any atomic formula [A(x1, x2, . . . , xn)] and sequence [(a1, a2, . . . , an)]of PA, we take SATCON(IS(ZF)) as:

[A(a1, a2, . . . , a

n)] is provable in ZF and, for any given

sequence [(b1, b2, . . . , bn)] of ZF, the formula [A(b1, b2, . . . , bn)]is decidable as either provable or not provable in ZF;

(d) we define the witness WZF as the meta-theory MZF of ZF whichcan always decide effectively whether or not SATCON(IS(ZF))holds in ZF.

Now, if the set-theoretic interpretation IP A(ZF, Cantor) of PA is sound, thenevery sound interpretation of ZF would, ipso facto, be a sound interpretation ofPA. In Appendix E, Section 12.2 I show, however, that this is not the case, andso the set-theoretic interpretation IP A(ZF, Cantor) of PA is not sound.

4.5 A purely algorithmic interpretation of PAWe finally consider the purely algorithmic interpretation IP A(N, Algorithmic) ofPA where:



(c) for any atomic formula [A(x1, x2, . . . , xn)] and sequence (a1, a2, . . . , an)ofN, we take SATCON(IP A(N)) as:

44cf. Godels Theorem XI in [Go31], p.36.

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A(a1, a2, . . . , a

n) holds in N and, for any given sequence

(b

1, b

2, . . . , b

n) ofN

, the proposition A

(b

1, b

2, . . . , b

n) isdecidable as either holding or not holding in N;

(d) we define the witness W(N, Algorithmic) as a Turing machine thatcomputes [A(x1, x2, . . . , xn)] and gives evidence that SATCON(IP A(N))is always effectively decidable in N:

Lemma 3 A(x1, x2, . . . , xn) is always algorithmically computable inN byW(N, Algorithmic).

Proof If [A(x1, x2, . . . , xn)] is an atomic formula of PA then, for any givensequence of numerals [b1, b2, . . . , bn], the PA formula [A(b1, b2, . . . , bn)] is anatomic formula of the form [c = d], where [c] and [d] are atomic PA formulasthat denote PA numerals. Since [c] and [d] are recursively defined formulas inthe language of PA, it follows from a standard result45 that, if PA is consistent,

then [c = d] is algorithmically computable as either true or false inN

. In otherwords, if PA is consistent, then [A(x1, x2, . . . , xn)] is algorithmically computable(since there is a Turing machine that computes [A(x1, x2, . . . , xn)] and, for anygiven sequence of numerals [b1, b2, . . . , bn], will halt with output 0 if [A(b1, b2,. . . , bn)] interprets as true in N; and halt with output 1 if [A(b1, b2, . . . , bn)]interprets as false in N). The lemma follows. 2

It follows that IP A(N, Algorithmic) is an algorithmic formulation of the stan-dard interpretation of PA in which we do not extraneously assume that Aris-totles particularisation holds over N.

We shall show that ifIP A(N, Algorithmic) is sound, then PA is not -consistent.Hence Aristotles particularisation does not hold over N, and the interpretation isfinitary and intuitionistically unobjectionable. Moreoversince the Law of theExcluded Middle is provable in an -inconsistent PA (and therefore holds in N)

it achieves this without the discomforting, stringent, Intuitionistic requirementthat we reject the underlying logic of PA!

5 Formally defining the standard interpretationof PA constructively

It follows from the analysis of the applicability of Tarskis inductive definitionsof satisfiability and truth in Section 4 that we can formally defineas detailedin Section 9, Appendix Bthe standard interpretation IP A(N, S tandard) of PAconstructively where:

(a) we define S as PA with standard first-order predicate calculus as

the underlying logic;


(c) we take SM(IP A(N, S tandard)) as a deterministic Turing machine.

We note that:

Theorem 2 The atomic formulas of PA are algorithmically verifiable under thestandard interpretationIP A(N, S tandard).

45For any natural numbers m, n, if m = n, then PA proves [(m = n)] ([Me64], p.110,Proposition 3.6). The converse is obviously true.

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Proof If [A(x1, x2, . . . , xn)] is an atomic formula of PA then, for any given

denumerable sequence of numerals [b1, b2, . . .], the PA formula [A(b1, b2, . . . , bn)]is an atomic formula of the form [c = d], where [c] and [d] are atomic PA formulasthat denote PA numerals. Since [c] and [d] are recursively defined formulas inthe language of PA, it follows from a standard result46 that, if PA is consistent,then [c = d] interprets as the proposition c = d which either holds or not for awitness WN in N.

Hence, if PA is consistent, then [A(x1, x2, . . . , xn)] is algorithmically verifi-able since, for any given denumerable sequence of numerals [b1, b2, . . .], we candefine a deterministic Turing machine that will compute [A(b1, b2, . . . , bn)] andhalt on null input as evidence that the PA formula [A(b1, b2, . . . , bn)] is decidableunder the interpretation.

The theorem follows. 2

It immediately follows that:

Corollary 1 The satisfaction and truth of PA formulas containing logicalconstants can be defined under the standard interpretation of PA in terms of theevidence provided by the computations of a deterministic Turing machine.

Corollary 2 The PA-formulas are decidable under the standard interpretationof PA if, and only if, they are algorithmically verifiable under the interpretation.

5.1 Defining algorithmic truth under the standard inter-pretation of PA

Now we note that, in addition to Theorem 2:

Theorem 3 The atomic formulas of PA are algorithmically computable underthe standard interpretationIP A(N, Standard).

Proof If [A(x1, x2, . . . , xn)] is an atomic formula of PA then we can definea deterministic Turing machine that will compute [A(x1, x2, . . . , xn)] and halton any given denumerable sequence of numerals [b1, b2, . . .] as evidence that thePA formula [A(b1, b2, . . . , bn)] is decidable under the interpretation.

The theorem follows. 2

This suggests the following definitions:

Definition 20 A well-formed formula [A] of PA is algorithmically true underIP A(N, S tandard) if, and only if, there is a deterministic Turing machine whichcomputes [A] and provides evidence that, given any denumerable sequence t ofN, t satisfies [A];

Definition 21 A well-formed formula [A] of PA is algorithmically false underIP A(N, S tandard) if, and only if, it is not algorithmically true under IP A(N).

46For any natural numbers m, n, if m = n, then PA proves [(m = n)] ([Me64], p.110,Proposition 3.6). The converse is obviously true.

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5.1.1 An algorithmic interpretation of the PA axioms

The significance of defining algorithmic truth under IP A(N, Standard) as aboveis that:

Lemma 4 The PA axioms PA1 to PA8 are algorithmically computable as algo-rithmically true overN under the interpretationIP A(N, Standard).

Proof Since [x + y], [x y], [x = y], [x] are defined recursively47, the PAaxioms PA1 to PA8 interpret as recursive relations that do not involve anyquantification. The lemma follows straightforwardly from Definitions 14 to 19in Section 4 and Theorem 2. 2

Lemma 5 For any given PA formula[F(x)], the Induction axiom schema [F(0)

(((x)(F(x) F(x

))) (x)F(x))] interprets as algorithmically true underIP A(N, S tandard).

Proof By Definitions 14 to 21:

(a) If [F(0)] interprets as algorithmically false underIP A(N, S tandard)the lemma is proved.

Since [F(0) (((x)(F(x) F(x))) (x)F(x))] interprets asalgorithmically true if, and only if, either [F(0)] interprets as algo-rithmically false or [((x)(F(x) F(x))) (x)F(x)] interpretsas algorithmically true.

(b) If [F(0)] interprets as algorithmically true and [(x)(F(x) F(x))] interprets as algorithmically false under IP A(N, S tandard), thelemma is proved.

(c) If [F(0)] and [(x)(F(x) F(x))] both interpret as algorith-mically true under IP A(N, S tandard), then by Definition 20 there is adeterministic Turing machine that computes [F(x)] and, for any nat-ural number n, will give evidence that the formula [F(n) F(n)]is true under IP A(N, S tandard).

Since [F(0)] interprets as algorithmically true underIP A(N, S tandard),it follows that there is a deterministic Turing machine that computes[F(x)] and, for any natural number n, will give evidence that the

formula [F(n)] is true under the interpretation.

Hence [(x)F(x)] is algorithmically true under IP A(N, S tandard).

Since the above cases are exhaustive, the lemma follows. 2

The Poincare-Hilbert debate: We note that Lemma 5 appears to settlethe Poincare-Hilbert debate48 in the latters favour. Poincare believed that theInduction Axiom could not be justified finitarily, as any such argument wouldnecessarily need to appeal to infinite induction. Hilbert believed that a finitaryproof of the consistency of PA was possible.

47cf. [Go31], p.17.48See [Hi27], p.472; also [Br13], p.59; [We27], p.482; [Pa71], p.502-503.

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Lemma 6 Generalisation preserves algorithmic truth under IP A(N, Standard).

Proof The two meta-assertions:

[F(x)] interprets as algorithmically true under IP A(N, S tandard)49

and

[(x)F(x)] interprets as algorithmically true under IP A(N, S tandard)

both mean:

[F(x)] is algorithmically computable as always true under IP A(N),Standard). 2

It is also straightforward to see that:

Lemma 7 Modus Ponens preserves algorithmic truth under IP A(N, S tandard).2

We thus have that:

Theorem 4 The axioms of PA are always algorithmically true under the inter-pretation IP A(N, S tandard), and the rules of inference of PA preserve the prop-erties of algorithmic satisfaction/truth underIP A(N, Standard)

50. 2

5.1.2 The algorithmic interpretation IP A(N, Algorithmic) of PA over N

We conclude that there is an algorithmic interpretation IP A(N, Algorithmic) of

PA over Nformally defined in Section 10, Appendix Csuch that:

Theorem 5 The interpretationIP A(N, Algorithmic) of PA is sound51.

Proof It follows immediately from Theorem 4 and Section 10, Appendix C, thatthe axioms of PA are always true under the interpretation IP A(N, Algorithmic),and the rules of inference of PA preserve the properties of satisfaction/truthunder IP A(N, Algorithmic). 2

We thus have a finitary proof that:

Theorem 6 PA is consistent. 2

6 A Provability Theorem for PAWe now show that PA can have no non-standard model 52, since it is algorith-mically complete in the sense that:

Theorem 7 (Provability Theorem for PA) A PA formula[F(x)] is PA-provableif, and only if, [F(x)] is algorithmically computable as always true inN.

49See Definition 1850Without appeal, moreover, to Aristotles particularisation.51In the sense of Definitions 9 and 10.52We consider the usual arguments for the existence of non-standard models of PA in Section

13, Appendix F.

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Proof We have by definition that [(x)F(x)] interprets as true under the inter-

pretation IP A(N, Algorithmic) if, and only if, [F(x)] is algorithmically computableas always true in N.

Since IP A(N, Algorithmic) is sound, it defines a finitary model of PA overNsay MP A()such that:

If [(x)F(x)] is PA-provable, then [F(x)] is algorithmically com-putable as always true in N;

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

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

(i) There is a finitary modelsay M1 of PA+[(x)F(x)] in which[F(x)] is algorithmically computable as always true in N.

(ii) There is a finitary modelsay M2of PA+[(x)F(x)] inwhich it is not the case that [F(x)] is algorithmically computableas always true in N.

The lemma follows. 2

Corollary 3 PA is categorical.

7 PA is not -consistent

In his seminal 1931 paper on formally undecidable arithmetical propositions53,Godel showed that54:

Lemma 8 If a Peano Arithmetic such as PA is -consistent, then there is aconstructively definable PA-formula [R(x)]55 such that neither [(x)R(x)) nor[(x)R(x)] are PA-provable56.2

Godel concluded that:

Lemma 9 Any-consistent Peano Arithmetic such as PA has a consistent, but-inconsistent, extension PA, obtained by adding the formula [(x)R(x)] asan axiom to PA57.2

Specifically, Godels reasoning shows that:

Lemma 10 If PA is consistent and [(x)R(x)] is assumed PA-provable, then[(x)R(x)] is PA-provable58.2

53[Go31].54[Go31], Theorem VI, p.24.55In his argument, Godel refers to this formula only by its Godel number r; [Go31], p.25,

Eqn.(12).56[Go31], p.25(1) & p.26(2).57[Go31], p.27.58This follows from Godels argument in [Go31], p.26(1).

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Lemma 11 If PA is -consistent and [(x)R(x)] is assumed PA-provable,

then [(x)R(x)] is PA-provable59

.2

However, by the argument in Theorem 7 it now follows that:

Corollary 4 The PA formula [(x)R(x)] is PA-provable. 2

Of course [(x)R(x)] interprets under IP A(N, Algorithmic) as the assertion:

There is no Turing machine that will compute [R(x)] and, for anygiven natural number n, provide evidence that R(n) is a true arith-metical proposition in N.

However, since Godel has shown that the PA-formula [R(n)] is PA-provable forany given PA-numeral [n], it follows that:

For any given natural number n, there is always some Turing ma-chine that will compute [R(n)] and halt on null input as evidencethat R(n) is a true arithmetical proposition in N.

Thus the PA-formula [(x)R(x)] is algorithmically verifiable as true over N, butnot algorithmically computable as true over N. The arithmetical relation R(x)is thus a Halting-type of relation, such that although R(x) is a tautology over N,there is no Turing machine that will compute [ R(x)] and, for any given naturalnumber n, give evidence that R(n) is a true arithmetical proposition in N.

We conclude that:

Corollary 5 PA is not -consistent.60

Proof Godel has shown that if PA is consistent, then [R(n)] is PA-provable forany given PA numeral [n]61. By Corollary 4 and the definition of-consistency,if PA is consistent then it is not -consistent. 2

Corollary 6 The standard interpretationIP A(N, Standard) of PA is not sound62,

and does not yield a model of PA63.

Proof By Corollary 8 if PA is consistent but not -consistent, then Aristotlesparticularisation does not hold over N. Since the standard, interpretation ofPA appeals to Aristotles particularisation, the lemma follows. 2

59This follows from Godels argument in [Go31], p.26(2).60This conclusion is contrary to accepted dogma. See, for instance, Davis remarks in

[Da82], p.129(iii) that . . . there is no equivocation. Either an adequate arithmetical logicis -inconsistent (in which case it is possible to prove false statements within it) or it hasan unsolvable decision problem and is subject to the limitations of Godels incompletenesstheorem.

61[Go31], p.26(2).62In the sense of Definitions 9 and 10.63I note that finitists of all huesranging from Brouwer [Br08] to Alexander Yessenin-

Volpin [He04]have persistently questioned the soundness of the standard interpretationIP A(N, S tandard).

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8 Appendix A: The significance of-consistency

and Hilberts programIn order to avoid intuitionistic objections to his reasoning in his seminal 1931paper on formally undecidable arithmetical propositions64, Kurt Godel did notassume that the standard interpretation of PA is sound65. Instead, Godel in-troduced the syntactic property of -consistency66 as an explicit assumptionin his formal reasoning67. Godel explained at some length68 that his reasonsfor introducing -consistency as an explicit assumption in his formal reasoningwas to avoid appealing to the semantic concept of classical arithmetical trutha concept which is implicitly based on an intuitionistically objectionable logicthat assumes Aristotles particularisation69 is valid over N.

However, we now show that if we assume the standard interpretation of PA

is sound

70

, then PA is consistent if, and only if, it is

-consistent.

8.0.3 Hilberts -Rule

To place the issue in the perspective of this paper, we consider the question:

Assuming that PA has a sound71 interpretation over N, is it truethat:

Algorithmic -Rule: If it is proved that the PA formula [F(x)]interprets as an arithmetical relation F(x) that is algorithmicallycomputable as true for any given natural number n, then the PAformula [(x)F(x)] can be admitted as an initial formula (axiom) inPA?

The significance of this query is that, as part of his program for givingmathematical reasoning a finitary foundation, Hilbert72 proposed an -Ruleas a finitary means of extending a Peano Arithmetic to a possible completion(i.e. to logically showing that, given any arithmetical proposition, either theproposition, or its negation, is formally provable from the axioms and rules ofinference of the extended Arithmetic).

Hilberts -Rule: If it is proved that the PA formula [F(x)] in-terprets as an arithmetical relation F(x) that is true for any givennatural number n, then the PA formula [(x)F(x)] can be admittedas an initial formula (axiom) in PA.

Now, in his 1931 paperwhich can, not unreasonably, be seen as the out-come of a presumed attempt to validate Hilberts -ruleGodel introduced theconcept of -consistency73, from which it follows that:

64[Go31].65In the sense of Definitions 9 and 10.66Definition 8.67[Go31], p.23 and p.28.68In his introduction on p.9 of [Go31].69Definition 1.70In the sense of Definitions 9 and 10.71In the sense of Definitions 9 and 10.72cf. [Hi30], pp.485-494.73[Go31], p.23.

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Lemma 12 If we meta-assume Hilberts -rule for PA, then a consistent PA

is necessarily -consistent74

.2

Proof If the PA formula [F(x)] interprets as an arithmetical relation F(x)that is true for any given natural number n, and the PA formula [(x)F(x)]can be admitted as an initial formula (axiom) in PA, [(x)F(x)] cannot bePA-provable if PA is consistent. The lemma follows. 2

Moreover, it follows from Godels 1931 paper that one consequence of assum-ing Hilberts -Rule is that there must, then, be an undecidable arithmeticalproposition75; a further consequence of which is that PA is essentially incom-plete.

However, since Godels argument in this paperfrom which he concludes theexistence of an undecidable arithmetical propositionis based on the weaker

(i.e., weaker than assuming Hilberts -rule) premise that a consistent PA canbe -consistent, the question arises whether an even weaker Algorithmic -Rule (which, prima facie, does not imply that a consistent PA is necessarily-consistent) can yield a finitary completion for PA as sought by Hilbert, albeitfor an -inconsistent PA.

8.0.4 Aristotles particularisation and -consistency

We shall now argue that these issues are related, and that placing them in anappropriate perspective requires questioning not only the persisting belief thatAristotles 2000-year old logic of predicatesa critical component of which isAristotles particularisationremains valid even when applied over an infinitedomain such as N, but also the basis of Brouwers denial of the Law of the

Excluded Middle following his challenge of the belief in 190876.Now, we have that:

Lemma 13 If PA is consistent but not -consistent, then there is some PA for-mula[F(x)] such that, under any sound77 interpretationsayIP A(N, Sound)ofPA overN:

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

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

Proof The lemma follows from the definition of-consistency and from Tarskis

standard definitions

78

of the satisfaction, and truth, of the formulas of a formalsystem such as PA under an interpretation as detailed in Section 4. 2

Further:

74However, we cannot similarly conclude from the the Algorithmic -Rule that a consistentPA is necessarily -consistent.

75Godel constructed an arithmetical proposition [R(x)] and showed that, if a Peano Arith-metic is -consistent, then both [(x)R(x)] and [(x)R(x)] are unprovable in the Arithmetic([Go31], p.25(1), p.26(2)).

76[Br08].77In the sense of Definitions 9 and 10.78[Ta33]; see also [Ho01] for an explanatory exposition. However, for standardisation and

convenience of expression, We follow the formal exposition of Tarskis definitions given in[Me64], p.50.

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Lemma 14 If the interpretationIP A(N) admits Aristotles particularisation overN79

, and the PA formula[(x)F(x)] interprets as true underIP A(N), then thereis some unspecified PA numeral [m] such that the PA formula [F(m)] interpretsas false underIP A(N).

Proof The lemma follows from Aristotles particularisation and Tarskis stan-dard definitions of the satisfaction, and truth, of the formulas of a formal systemsuch as PA under an interpretation. 2

Hence:

Lemma 15 If PA is consistent and Aristotles particularisation holds over N,there can be no PA formula [F(x)] such that, under any sound80 interpretationIP A(N, Sound) of PA overN:

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

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

Proof The lemma follows from the previous two lemma. 2

In other words81:

Corollary 7 If PA is consistent and Aristotles particularisation holds overN,then PA is -consistent. 2

It follows that:

Lemma 16 If Aristotles particularisation holds overN, then PA is consistentif, and only if, it is -consistent.

Proof If PA is -consistent then, since [n = n] is PA-provable for any given PAnumeral [n], we cannot have that [(x)(x = x)] is PA-provable. Since an in-consistent PA proves [(x)(x = x)], an -consistent PA cannot be inconsistent.2

The arguments of this section thus suggest that J. Barkley Rossers extension ofGodels argument82 succeeds in avoiding an explicit assumption of -consistencyonly by implicitly appealing to Aristotles particularisation.

It further follows that:

Corollary 8 If PA is consistent but not -consistent, then Aristotles particu-larisation does not hold overN. 2

As the classical, standard, interpretation of PAsay IP A(N, S tandard)appeals to Aristotles particularisation83, it follows that:

Corollary 9 If PA is consistent but not -consistent, then the standard inter-pretation IP A(N, S tandard) of PA is not sound

84, and does not yield a model ofPA. 2

79As, for instance, in [Me64], pp.51-52 V(ii).80In the sense of Definitions 9 and 10.81The above argument is made explicit in view of Martin Davis remark in [Da82], p.129,

that such a proof of -consistency may be . . . open to the objection of circularity.82[Ro36].83See, for instance, [Me64], p.107 and p.52(V)(ii).84In the sense of Definitions 9 and 10.

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9 Appendix B: The standard interpretation

IP A(N, Standard) of PA over NWe define the satisfiability and truth of the formulas of PA under the standardinterpretation IP A(N, S tandard) of PA over N formally as follows:

Definition 22 If [A] is an atomic formula [A(x1, x2, . . . , xn)] of PA, then thedenumerable sequence (a1, a2, . . .) in the domainN of the interpretationIP A(N,Standard) of PA satisfies [A] if, and only if:

(i) [A(x1, x2, . . . , xn)] interprets underIP A(N, Standard) as a uniquerelation A(x1, x2, . . . , xn) inN for any witness WN ofN;

(ii) for any atomic formula [A(x1, x2, . . . , xn)] of PA, and any given

denumerable sequence (b1, b2, . . .) ofN

, there is a deterministic Tur-ing Machine that computes A(b1, b2, . . . , bn) and halts on null inputyielding objective evidence by which any witness WN ofN candefinewhether the proposition A(b1, b2, . . . , bn) holds or not inN;

(iii) A(a1, a2, . . . , an) holds inN for anyWN.

We inductively assign truth values of satisfaction, truth, and falsity tothe compound formulas of PA under the interpretation IP A(N, S tandard) in termsof only the satisfiability of the atomic formulas of PA over N as follows85:

Definition 23 A denumerable sequence s ofN satisfies [A] underIP A(N, Standard) if, and only if, s does not satisfy [A];

Definition 24 A denumerable sequence s ofN satisfies [A B] underIP A(N,Standard) if, and only if, either it is not the case that s satisfies [A], ors satisfies[B];

Definition 25 A denumerable sequence s ofN satisfies [(xi)A] underIP A(N,Standard) if, and only if, given any denumerable sequence t of N which differsfrom s in at most the ith component, t satisfies [A];

Definition 26 A well-formed formula [A] ofN is true86 underIP A(N, S tandard)if, and only if, given any denumerable sequence t ofN, t satisfies [A];

Definition 27 A well-formed formula [A] ofN is false underIP A(N, S tandard)if, and only if, it is not the case that [A] is true underIP A(N, Standard).

10 Appendix C: The algorithmic interpretationIP A(N, Algorithmic) of PA over N

We define the satisfiability and truth of the formulas of PA under the algo-rithmic interpretation IP A(N, Algorithmic) of PA over N formally as follows:

85Compare [Me64], p.51; [Mu91].86Note that this definition of truth is best described as instantiational when compared

to the corresponding algorithmic definition of truth (Definition 32) in Section 10. The sig-nificance of the distinction between instantiational and algorithmic methods is highlightedin Section 11, Appendix D.

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Definition 28 If [A] is an atomic formula [A(x1, x2, . . . , xn)] of PA, then the

denumerable sequence (a1, a2, . . .) in the domainN

of the interpretationIP A(N,Algorithmic) of PA satisfies [A] if, and only if:

(i) [A(x1, x2, . . . , xn)] interprets underIP A(N, Algorithmic) as a uniquerelation A(x1, x2, . . . , xn) inN for any witness WN ofN;

(ii) for any atomic formula [A(x1, x2, . . . , xn)] of PA, there is a de-terministic Turing Machine that computes [A(x1, x2, . . . , xn)] andhalts on any given denumerable sequence (b1, b2, . . .) ofN, yieldingobjective evidence by which any witness WN ofN candefine whetherthe proposition A(b1, b2, . . . , bn) holds or not inN;

(iii) A(a1, a2, . . . , an) holds inN for anyWN.

We inductively assign truth values of satisfaction, truth, and falsity tothe compound formulas of PA under the interpretation IP A(N, Algorithmic) interms of only the satisfiability of the atomic formulas of PA over N as follows87:

Definition 29 A denumerable sequence s ofN satisfies [A] underIP A(N, Algorithmic) if, and only if, s does not satisfy [A];

Definition 30 A denumerable sequence s ofN satisfies [A B] underIP A(N,Algorithmic) if, and only if, either it is not the case that s satisfies [A], or ssatisfies [B];

Definition 31 A denumerable sequence s ofN satisfies [(xi)A] underIP A(N,Algorithmic) if, and only if, given any denumerable sequence t ofN which differs

from s in at most the ith component, t satisfies [A].

Definition 32 A well-formed formula[A] ofN is true88 underIP A(N, Algorithmic)if, and only if, given any denumerable sequence t ofN, t satisfies [A];

Definition 33 A well-formed formula[A] ofN is false underIP A(N, Algorithmic)if, and only if, it is not the case that [A] is true underIP A(N, Algorithmic).

11 Appendix D: The need for explicitly distin-guishing between instantiational and uni-form methods

It is significant that both Kurt Godel (initially) and Alonzo Church (subseque-ntlypossibly under the influence of Godels disquietitude) enunciated Churchsformulation of effective computability as a Thesis because Godel was instinc-tively uncomfortable with accepting it as a definition that minimally capturesthe essence of intuitive effective computability89.

Godels reservations seem vindicated if we accept that a number-theoreticfunction can be effectively computable instantiationally (in the sense of being

87Compare [Me64], p.51; [Mu91].88Algorithmically. The significance of the distinction between instantiaional and algorith-

mic methods is highlighted in Section 11, Appendix D.89See [Si97].

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algorithmically verifiable as envisaged in Definition 12 above), but not by a

uniform method (in the sense of being algorithmically computable as envisagedin Definition 13).The significance of the fact (considered above in Section 4) that truth too

can be effectively decidable both instantiationally and by a uniform (algorithmic)method under the standard interpretation of PA is reflected in Godels famous1951 Gibbs lecture90, 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 case

if the proposition in question were some equation F(n) = G(n) oftwo number-theoretical functions which could be verified up to verygreat numbers n.91

Such a possibility is also implicit in Turings remarks92:

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 of8 that P is not computable. It is (so far aswe 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 method

is necessary in order to obtain more figures.

The need for placing such a distinction on a formal basis has also beenexpressed explicitly on occasion93. Thus, Boolos, Burgess and Jeffrey94 definea diagonal function, d, any value of which can be decided 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 effectivelycomputable instantiationally but not algorithmically. Yet, following Church andTuring, such functions are labeled as uncomputable95!

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

90[Go51].91Parikhs paper [Pa71] can also be viewed as an attempt to investigate the consequences

of expressing the essence of Godels remarks formally.92[Tu36], 9(II), p.139.93Parikhs distinction between decidability and feasibility in [Pa71] also appears to echo

the need for such a distinction.94[BBJ03], p. 37.95The 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 threesenses, without making explicit distinctions between them (see [?]).

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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 integernif we dont run out of time.96

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 97ascomputable, on the grounds that the time needed to compute it increases mono-tonically with n, is curious98; the same applies to any total Turing-computablefunction f(n)99!

12 Appendix E: No model of PA can admit atransfinite ordinal

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

[x = 0 (y)(x = y)]

This translates, under every unrelativised interpretation of PA, as:

If x denotes an element in the domain of an unrelativised interpretation ofPA, either x is 0, or x is a successor.

Further, in every such interpretation of PA, ifG(x) denotes the interpretationof [G(x)]:

(a) G(0) is true;(b) If G(x) is true, then G(x) is true.

Hence, by Godels completeness theorem:

(c) PA proves [G(0)];(d) PA proves [G(x) G(x)].

Godels Completeness Theorem: In any first-order predicate calculus, the theo-rems are precisely the logically valid well-formed formulas (i. e. those that aretrue in every model of the calculus).

Further, by Generalisation:

(e) PA proves [(x)(G(x) G(x))];

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

Hence, by Induction:

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

Induction Axiom Schema of PA: For any formula [F(x)] of PA:[F(0) ((x)(F(x) F(x)) (x)F(x))]

96[BBJ03], p.37.97Chaitins 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].

98The incongruity of this is addressed by Parikh in [Pa71].99The 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 of n,where the length of the program will, sometime, reference n.

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In other words, except 0, every element in the domain of any unrelativised

interpretation of PA is a successor. Further, x can only be a successor of aunique element in any such interpretation of PA.

12.1 PA and Ordinal Arithmetic have no common model

Now, since Cantors first limit ordinal, , is not the successor of any ordinal inthe sense required by the PA axioms, and since there are no infinitely descendingsequences of ordinals100 in a modelif anyof set-theory, PA and OrdinalArithmetic101 cannot have a common model, and so we cannot consistentlyextend PA to OA simply by the addition of more axioms.

12.2 Why PA has no set-theoretical model

We can define the usual order relation


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13 Appendix F: Why the usual arguments for a

non-standard model of PA are unconvincingAlthough we can define a model of Arithmetic with an infinite descending se-quence of elements102, any such model is isomorphic to the true arithmetic103of the integers (negative plus positive), and not to any model of PA104.

Moreoveras we show in the next sectionwe cannot assume that we canconsistently add a constant c to PA, along with the denumerable axioms [(c =0)], [(c = 1)], [(c = 2)], . . . , since this would presume that which is soughtto be proven, viz., that PA has a non-standard model.

We cannot thereforeas suggested in standard texts105apply the Com-pactness Theorem and the (upward) Lowen-heim-Skolem Theorem to concludethat PA has a non-standard model!

Compactness Theorem: If every finite subset of a set of sentences has a model,then the whole set has a model106.

Upward Lowenheim-Skolem Theorem: Any set of sentences that has an infinitemodel has a non-denumerable model107.

13.1 A formal argument for a non-standard model of PA

The following argument108 attempts to validate the above line of reasoningsuggested by standard texts for the existence of non-standard models of PA:

1. Let be the structure that serves to define a sound interpretat-ion of PA, say [N].

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

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

4. Let be the countable set of all PA-formulas of the form [ cn = (cn+1)],

where the index n is a natural number.

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

6. T[N] plus any finite set of members of has a model, e.g., [N] itself,

since [N] is a model of any finite descending chain of successors.

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

8. M has an infinite descending sequence with respect to because it is a

102eg. [BBJ03], Section 25. 1, p303.103[BBJ03]. p150. Ex. 12. 9.104[BBJ03]. Corollary 25. 3, p306.105eg. [BBJ03]. p306; [Me64], p112, Ex. 2.106[BBJ03]. p147.107[BBJ03]. p163.108[Ln08].

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model of .

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

Now, ifas claimed above[N] is a model of T[N] plus any finite set ofmembers of , then all PA-formulas of the form [cn = (cn+1)

] are PA-provable, is a proper sub-set of the PA-provable formulas, and T is identically T[ N].

(The argument cannot be that some PA-formula of the form [cn = (cn+1)] is

true in [N], but not PA-provable, as this would imply that PA+[(cn = (cn+1))]

has a model other than [N ]; in other words, it would presume that PA has anon-standard model.109)

Consequently, the postulated model M of T in (7), by Compactness, is themodel [N] that defines T[N]. However, [N] has no infinite descending sequencewith respect to , even though it is a model of . Hence the argument does not

establish the existence of a non-standard model of PA with an infinite descendingsequence with respect to the successor function .

13.2 The (upward) Skolem-Lowenheim theorem appliesonly to first-order theories that admit an axiom ofinfinity

We note, moreover, that the non-existence of non-standard models of PA wouldnot contradict the (upward) Skolem-Lowenheim theorem, since the proof ofthis theorem implicitly limits its applicability amongst first-order theories tothose that are consistent with an axiom of infinityin the sense that the proofimplicitly requires that a constant, say c, along with a denumerable set of axioms

to the effect that c = 0, c = 1, . . ., can be consistently added to the theory.However, as seen in the previous section, this is not the case with PA.

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

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