The standard style of argument used to prove that a theory is unde- cidable relies on certain consistency assumptions, usually that some fragment or other is negation consistent. In a non-paraconsistent set- ting, this amounts to an assumption that the theory is non-trivial, but these diverge when theories are couched in paraconsistent logics. Furthermore, there are general methods for constructing inconsistent models of arithmetic from consistent models, and the theories of such inconsistent models seem likely to differ (...) in terms of complexity. In this paper, I begin to explore this terrain, working, particularly, in incon- sistent theories of arithmetic couched in three-valued paraconsistent logics which have strong (i.e. detaching) conditionals. (shrink)

This paper presents proof that Buss's S22 can prove the consistency of a fragment of Cook and Urquhart's PV from which induction has been removed but substitution has been retained. This result improves Beckmann's result, which proves the consistency of such a system without substitution in bounded arithmetic S12. Our proof relies on the notion of "computation" of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and (...) either its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven. This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss's S12. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies. (shrink)

A proof of the consistency of Heyting arithmetic formulated in natural deduction is given. The proof is a reduction procedure for derivations of falsity and a vector assignment, such that each reduction reduces the vector. By an interpretation of the expressions of the vectors as ordinals each derivation of falsity is assigned an ordinal less than ε 0, thus proving termination of the procedure.

We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of IΔ0 whose Herbrand Consistency is not provable in IΔ0. We also show the existence of an IΔ0-derivable Π1-sentence such that IΔ0 cannot prove its Herbrand Consistency.

Gödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, Fundamenta Mathematical vol. 171 (2002), pp. 279-292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories I∆₀+ Ωm, with m ≥ 2, any witness for (...) any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory T ⊇ I∆₀ + Ω₂ in t itself. In this paper, the above results are generalized for I∆₀ + Ω₁. Also after tailoring the definition of Herbrand consistency for I∆₀ we prove the corresponding theorems for I∆₀. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories I∆₀ + Ω₁ and I∆₀. (shrink)

This paper offers an elementary proof that formal arithmetic is consistent. The system that will be proved consistent is a first-order theory R♯, based as usual on the Peano postulates and the recursion equations for + and ×. However, the reasoning will apply to any axiomatizable extension of R♯ got by adding classical arithmetical truths. Moreover, it will continue to apply through a large range of variation of the un- derlying logic of R♯, while on a simple (...) and straightforward translation, the classical first-order theory P♯ of Peano arithmetic turns out to be an exact subsystem of R♯. Since the reasoning is elementary, it is formalizable within R♯ itself; i.e., we can actually demonstrate within R♯ (or within P♯, if we care) a statement that, in a natural fashion, asserts the consistency of R♯ itself. The reader is unlikely to have missed the significance of the remarks just made. In plain English, this paper repeals Goedel’s famous second theorem. (That’s the one that asserts that sufficiently strong systems are inadequate to demonstrate their own consistency.) That theorem (or at least the significance usually claimed for it) was a mis- take—a subtle and understandable mistake, perhaps, but a mistake nonetheless. Accordingly, this paper reinstates the formal program which is often taken to have been blasted away by Goedel’s theorems— namely, the Hilbert program of demonstrating, by methods that everybody can recognize as effective and finitary, that intuitive mathematics is reliable. Indeed, the present consistency proof for arithmetic will be recognized as correct by anyone who can count to 3. (So much, indeed, for the claim that the reliability of arithmetic rests on transfinite induction up to ε0, and for the incredible mythology that underlies it.). (shrink)

This volume contains six new and fifteen previously published essays -- plus a new introduction -- by Storrs McCall. Some of the essays were written in collaboration with E. J. Lowe of Durham University. The essays discuss controversial topics in logic, action theory, determinism and indeterminism, and the nature of human choice and decision. Some construct a modern up-to-date version of Aristotle's bouleusis, practical deliberation. This process of practical deliberation is shown to be indeterministic but highly controlled and the antithesis (...) of chance. Others deal with the concept of branching four-dimensional space-time, explain non-local influences in quantum mechanics, or reconcile God's omniscience with human free will. The eponymous first essay contains the proof of a fact that in 1931 Kurt Gödel had claimed to be unprovable, namely that the set of arithmetic truths forms a consistent system. (shrink)

When a structure or class of structures admits an unbounded induction, we can do arithmetic on the stages of that induction: if only bounded inductions are admitted, then clearly each inductively definable relation can be defined using a finite explicit expression. Is the converse true? We examine evidence that the converse is true, in positive elementary induction . We present a stronger conjecture involving the language L consisting of all L∞ω formulas with a finite number of variables, and examine (...) a combinatorial property equivalent to “all L-definable relations are elementary”. (shrink)

It is argued that the goal of Hilbert's program was to prove the model-theoretical consistency of different axiom systems. This Hilbert proposed to do by proving the deductive consistency of the relevant systems. In the extended independence-friendly logic there is a complete proof method for the contradictory negations of independence-friendly sentences, so the existence of a single proposition that is not disprovable from arithmetic axioms can be shown formally in the extended independence-friendly logic. It can also be proved by (...) means of independence-friendly logic that proof-theoretical consistency of a sentence S implies the existence of a model in which S is not false. Hence the consistency of the axioms of arithmetic in the sense of being not-false in a model can be proved. (shrink)

We prove that the Gödel incompleteness theorem holds for a weak arithmetic T = IΔ0 + Ω2 in the form where Cons H (T) is an arithmetic formula expressing the consistency of T with respect to the Herbrand notion of provability.

We consider equational theories based on axioms for recursively defining functions, with rules for equality and substitution, but no form of induction—we denote such equational theories as PETS for pure equational theories with substitution. An example is Cook’s system PV without its rule for induction. We show that the Bounded Arithmetic theory $\mathrm {S}^{1}_2$ proves the consistency of PETS. Our approach employs model-theoretic constructions for PETS based on approximate values resembling notions from domain theory in Bounded Arithmetic, which (...) may be of independent interest. (shrink)

The 1-consistency of arithmetic is shown to be equivalent to the existence of fixed points of a certain type of update procedure, which is implicit in the epsilon-substitution method.

What Gödel referred to as “outer” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of Hilbert’s finitist (...) ideal which is achieved by going back to Kronecker’s programme of a general arithmetic of forms or homogeneous polynomials. The paper can be seen as a historical complement to our result on “The Internal Consistency of Arithmetic with Infinite Descent” . An internal consistency proof for arithmetic means that transfinite induction is not needed and that arithmetic can be shown to be consistent within the bounds of arithmetic, that is with the help of Fermat’s infinite descent and Kronecker’s general or polynomial arithmetic, thus returning into arithmetic without the detour of Cantor’s transfinite arithmetic of ideal elements. (shrink)

We define and study a new restricted consistency notion RCon ∗ for bounded arithmetic theories T 2 j . It is the strongest ∀ Π 1 b -statement over S 2 1 provable in T 2 j , similar to Con in Krajíček and Pudlák, 29) or RCon in Krajı́ček and Takeuti 107). The advantage of our notion over the others is that RCon ∗ can directly be used to construct models of T 2 j . We apply this (...) by proving preservation theorems for theories of bounded arithmetic of the following well-known kind: The ∀ Π 1 b -separation of bounded arithmetic theories S 2 i from T 2 j is equivalent to the existence of a model of S 2 i which does not have a Δ 0 b -elementary extension to a model of T 2 j . More specific, let M ⊨ Ω 1 nst denote that there is a nonstandard element c in M such that the function n ↦ 2 log c is total in M . Let BLΣ 1 b be the bounded collection schema for Σ 1 b -formulas. We obtain the following preservation results: the ∀ Π 1 b -separation of S 2 i from T 2 j is equivalent to the existence of 1. a model of S 2 i +Ω 1 nst which is 1 b -closed w.r.t. T 2 j , 2. a countable model of S 2 i + BLΣ 1 b without weak end extensions to models of T 2 j. (shrink)

The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that it is trivial that relevant arithmetic is absolutely consistent, but classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under, I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is (...) exactly the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under, I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment. (shrink)

The lack of conceptual analysis within cognitive science results in multiple models of the same phenomena. However, these models incorporate assumptions that contradict basic structural features of the domain they are describing. This is particularly true about the domain of mathematical cognition. In this paper we argue that foundational theoretic aspects of psychological models for language and arithmetic should be clarified before postulating such models. We propose a means to clarify these foundational concepts by analyzing the distinctions between metric (...) and linguistic compositionality, which we use to assess current models of mathematical cognition. Our proposal is consistent with the scientific methodology that determines that careful conceptual analysis should precede theoretical descriptions of data. 2012 APA, all rights reserved). (shrink)

This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of (...) this paper are: (i) there is a consistent extension of the hyperarithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, and (ii) the hyperarithmetic fragment of Hume's Principle does not interpret the hyperarithmetic fragment of second-order Peano arithmetic, so that in this specific sense there is no predicative version of Frege's Theorem. (shrink)

An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯, the closure of the Peano axioms in Wansing’s connexive logic C. Namely, the paper conjectured that C♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and (...) Nelson’s paraconsistent N4, this also supplements Nelson’s own proof of the Post consistency of N4♯. Insofar as the present technique allows infinite models, this resolves Nelson’s concern that N4♯ is of interest only to those accepting that there are finitely many natural numbers. (shrink)

Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by Maietti and Sambin From sets and types to topology and analysis: practicable foundations for constructive mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for short MF, (...) completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory à la Martin-Löf, called mTT. The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church’s thesis of a fragment of intensional Martin-Löf’s type theory, called \, where mTT can be easily interpreted. Then to show the consistency of \ with CT we interpret it within Feferman’s predicative theory of non-iterative fixpoints \ by extending the well known Kleene’s realizability semantics of intuitionistic arithmetics so that CT is trivially validated. More in detail the fragment \ we interpret consists of first order intensional Martin-Löf’s type theory with one universe and with explicit substitution rules in place of usual equality rules preserving type constructors -rule which is not valid in our realizability semantics). A key difficulty encountered in our interpretation was to use the right interpretation of lambda abstraction in the applicative structure of natural numbers in order to model all the equality rules of \ correctly. In particular the universe of \ is modelled by means of \-fixpoints following a technique due first to Aczel and used by Feferman and Beeson. (shrink)

Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially (...) shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it relevant also for the epistemology of mathematics. (shrink)

Gödel’s Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer’s floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem and techniques for evading it.

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the (...) present such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested. (shrink)

The incompleteness of formal systems for arithmetic has been a recognized fact of mathematics. The term “incompleteness” suggests that the formal system in question fails to offer a deduction which it ought to. This chapter focuses on the status of a formal system, Peano Arithmetic, and explores a viewpoint on which Peano Arithmetic occupies an intrinsic, conceptually well-defined region of arithmetical truth. The idea is that it consists of those truths which can be perceived directly from the (...) purely arithmetical content of a categorical conceptual analysis of the notion of natural number. The chapter explores its conceptual stability in light of the apparently destabilizing effect, through extensibility, of the phenomenon of incompleteness. It also offers heuristic and conceptual support for the viewpoint that Peano Arithmetic is the strongest natural first-order system for arithmetic, and that there is a sense in which it is complete with respect to purely arithmetical truth. (shrink)

General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity (...) and Associativity of Addition and Multiplication, but also Lagrange’s Four-Square Theorem. Adding one more axiom, the one-oneness of succession, one can prove many more theorems, such as Quadratic Reciprocity and Fermat’s Little Theorem. By looking at arithmetic in this general setting, one receives a deeper understanding of the underlying structures. (shrink)

This article presents an axiom system for an arithmetic of the even and the odd, one that is stronger than those discussed in Pambuccian (2016) and Menn & Pambuccian (2016). It consists of universal sentences in a language extending the usual one with 0, 1, +, ·, <, – with the integer part of the half function$[{ \cdot \over 2}]$, and two unary operation symbols.

We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \, there exists a \ numeration \\) of T such that the provability logic of the provability predicate \\) naturally constructed from \\) is exactly \ \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.

Solovay proved the arithmetical completeness theorem for the system GL of propositional modal logic of provability. Montagna proved that this completeness does not hold for a natural extension QGL of GL to the predicate modal logic. Let Th(QGL) be the set of all theorems of QGL, Fr(QGL) be the set of all formulas valid in all transitive and conversely well-founded Kripke frames, and let PL(T) be the set of all predicate modal formulas provable in Tfor any arithmetical interpretation. Montagna’s results (...) are described as Th(QGL) ⊊ (Fr(QGL), PL(PA) ⊈ Fr(QGL), and Th(QGL) ⊊ PL(PA). In this paper, we prove the following three theorems: (1) Fr(QGL) ⊈ PL(T) for any Σ1-sound recursively enumerable extension T of I Σ1, (2) PL(T) ⊈ Fr(QGL) for any recursively enumerable S1755020312000275_inline1 A -theory T extending I Σ1, and (3) Th(QGL) ⊊ Fr(QGL) ∩ PL(T) for any recursively enumerable S1755020312000275_inline2 A -theory T extending I Σ2. To prove these theorems, we use iterated consistency assertions and nonstandard models of arithmetic, and we improve Artemov’s lemma which is used to prove Vardanyan’s theorem on the Π0 2-completeness of PL(T). (shrink)

We prove that for any recursively axiomatized consistent extension T of Peano Arithmetic, there exists a \ provability predicate of T whose provability logic is precisely the modal logic \. For this purpose, we introduce a new bimodal logic \, and prove the Kripke completeness theorem and the uniform arithmetical completeness theorem for \.

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite (...) sets and series and quantum information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself. (shrink)

Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive evidence (...) in arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses. (shrink)

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few (...) worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture. (shrink)

Our last big theorem – Theorem 6 – tells us that if a theory meets certain conditions, then it must be negation incomplete. And we made some initial arm-waving remarks to the effect that it seems plausible that we should want theories which meet those conditions. Later, we announced that there actually is a consistent weak arithmetic with a first-order logic which meets the conditions (in which case, stronger arithmetics will also meet the conditions); but we didn’t say (...) anything about what such a weak theory really looks like. In fact, we haven’t looked at any detailed theory of arithmetic yet! It is high time, then, that we stop operating at the extreme level of abstraction of Episodes 1 and 2, and start getting our hands dirty. This Episode introduces a couple of weak arithmetics, before we tackle the canonical first-order arithmetic PA in the following instalment. By all means skip fairly lightly over some of the more boring proof details! But it is certainly worth getting just a flavour of how these two simple formal theories work. (shrink)

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell's Paradox being derivable in it. This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege's Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson (...) arithmetic, Q, is relatively interpretable in the simple predicative fragment. The philosophical significance of the result is discussed. (shrink)

Arithmetical cognition is the result of enculturation. On a personal level of analysis, enculturation is a process of structured cultural learning that leads to the acquisition of evolutionarily recent, socio-culturally shaped arithmetical practices. On a sub-personal level, enculturation is realized by learning driven plasticity and learning driven bodily adaptability, which leads to the emergence of new neural circuitry and bodily action patterns. While learning driven plasticity in the case of arithmetical practices is not consistent with modularist theories of mental (...) architecture, it can be enriched by the theory of neural reuse. According to neural reuse, cerebral regions are reused to contribute to multiple neural circuits in functionally constrained ways throughout ontogeny. By hypothesis, learning driven plasticity is complemented by learning driven bodily adaptability, which suggests that there is an interesting functional relationship between finger gnosis, finger counting, and arithmetical practices. The emerging perspective on enculturated arithmetical cognition will be complemented by considerations on associated developmental and acquired disorders, namely developmental dyscalculia and acquired acalculia. The upshot is that we need to take the cerebral, extra-cerebral bodily, and socio-cultural dimensions of enculturation into account in order to arrive at a better understanding of the phylogenetic and ontogenetic conditions of arithmetical cognition. (shrink)

Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We will show here that it is consistent that the monadic second-order theory of no chain interprets Peano Arithmetic.

On the one hand, first-order theories are able to assert the existence of objects. For instance, ZF set theory asserts the existence of objects called the power set, while Peano Arithmetic asserts the existence of zero. On the other hand, a first-order theory may or not be consistent: it is if and only if no contradiction is a theorem. Let us ask, What is the connection between consistency and existence?

SummaryAccording to Dummett's understanding of Frege, the sense of a denoting expression is a procedure for determining its denotation. The purpose of this article is to pursue this suggestion and develop a semi‐formal interpretation of Fregean sense for the special case of a first‐order language of arithmetic. In particular, we define the sense of each arithmetic expression to be a hypothetical process to determine the denoted number or truth value. The sense‐process is “hypothetical” in that the senses of (...) some expressions cannot be completed in a finite number of steps. The motivation for this work is to explicate an intensional notion of effectiveness in terms of the senses of function names. It is proposed, in particular, that an expression is effective if and only if its sense can be completed in a finite number of steps. It is shown that a function is computable if and only if it has an effective name.RésuméSelon la lecture que Dummet donne de Frege, le sens ?on;une expression consiste en une procédure pour déterminer sa dénotation. Le but de cet article est de développer cette suggestion par une interprétation semi‐formelle du sens frégéen dans le cas particulier ?on;un langage de premier ordre en arithmétique. En particulier, nous définissons comme sens de toute expression mathématique un procédé hypothétique pour déterminer le nombre dénoté ou la valeur de vérité. Le procédé créateur de sens est hypothetique en ce sens que les sens de certaines expressions ne peuvent pas être donnés en un nombre fini de pas. Le but de cette étude est ?on;expliciter une notion intensionnelle ?on;effectuabilite en termes de sens des noms de fonction. II est en particulier propose qu'une expression soit dite effectuable si et seulement si son sens peut être épuisé en un nombre fini de pas. On montre qu'une fonction est calculable si et seulement si elle a un nom effectuable.ZusammenfassungDummets Auffassung von Frege gemäss besteht der Sinn eines denotierenden Ausdruckes in einem Verfahren, das dessen Denotation bestimmt. Es ist der Zweck der vorliegenden Arbeit, die‐sen Vorschlag weiterzufuUhren und eine halbformale Interpretation des Fregeschen Sinnes fiir den besonderen Fall der Sprache erster Stufe der Arithmetik zu entwickeln. Es wird im besonderen der Sinn eines jeden arithmetischen Ausdruckes als hypothetisches Verfahren zur Bestimmung der denotierten Zahl oder des Wahrheitswertes definiert. Die Sinngebung ist insofern hypothe‐tisch, als der Sinn gewisser Ausdrucke nicht in einer endlichen Zahl von Schritten gegeben wer‐den kann. Das Anliegen der Arbeit grundet in der Absicht, einen intensionalen Begriff der Effek‐tivitat aufgrund des Sinnes von Funktionsnamen zu erlautern. Es wird im besonderen festgelegt, dass ein Ausdruck dann und nur dann effektiv sei, wenn sein Sinn in einer endlichen Zahl von Schritten erreicht werden kann. Es wird gezeigt, dass eine Funktion genau dann berechenbar ist, wenn sie einen effektiven Namen hat. (shrink)

This present collection of (translations of) reviews is intended to help obtain a more balanced picture of the reception and impact of Edmund Husserl’s first book, the 1891 Philosophy of Arithmetic. One of the insights to be gained from this non-exhaustive collection of reviews is that the Philosophy of Arithmetic had a much more widespread reception than hitherto assumed: in the present collection alone there already are fourteen, all published between 1891 and 1895. Three of the reviews appeared (...) in mathematical journals (Jahrbuch über die Fortschritte der Mathematik, Zeitschrift für Mathematik und Physik, and Zeitschrift für mathema- tischen und naturwissenschaftlichen Unterricht), three were published in English journals (The Philosophical Review, The Monist, Mind), two were written by other members of the School of Brentano (Franz Hillebrand and Alois Höfler). Some of the reviews and notices appear to be very superficial, consisting merely of para- phrases (often without references) and lists of topics taken from the table of con- tents, presenting barely acceptable summaries. Others, among which Höfler might be the most significant, engage much more deeply with the topics and problems that Husserl addresses, analyzing his approach in the context of the mathematics of his time and the School of Brentano. (shrink)

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+1-formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first-order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, (...) we obtain results on the quantifier complexity, finite axiomatizability and relative strength of schemes for Δn+1-formulas. (shrink)

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6]. In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result (...) used as a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3, which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli. We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete. In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest. (shrink)

PG (Plural Grundgesetze) is a predicative monadic second-order system which is aimed to derive second-order Peano arithmetic. It exploits the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. In this paper, a model-theoretical consistency proof for the system PG is provided.

We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of (...) models for these systems. In both cases we give a consistency proof, but naturally we have to assume more than the mere comprehension axioms. (shrink)

Axiomatic set theory with full comprehension is known to be consistent in Łukasiewicz fuzzy predicate logic. But we cannot assume the existence of natural numbers satisfying a simple schema of induction; this extension is shown to be inconsistent.

In this paper, we show that the predicate logics of consistent extensions of Heyting's Arithmetic plus Church's Thesis with uniqueness condition are complete $\Pi _{2}^{0}$. Similarly, we show that the predicate logic of HA*, i.e. Heyting's Arithmetic plus the Completeness Principle (for HA*) is complete $\Pi _{2}^{0}$. These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko's method to use Tennenbaum's Theorem to prove 'categoricity of interpretations' under certain assumptions.

An epistemic formalization of arithmetic is constructed in which certain non-trivial metatheoretical inferences about the system itself can be made. These inferences involve the notion of provability in principle, and cannot be made in any consistent extensions of Stewart Shapiro's system of epistemic arithmetic. The system constructed in the paper can be given a modal-structural interpretation.

This paper gives a Gentzen-style proof of the consistency of Heyting arithmetic in an intuitionistic sequent calculus with explicit rules of weakening, contraction and cut. The reductions of the proof, which transform derivations of a contradiction into less complex derivations, are based on a method for direct cut-elimination without the use of multicut. This method treats contractions by tracing up from contracted cut formulas to the places in the derivation where each occurrence was first introduced. Thereby, Gentzen’s heightline argument, (...) which introduces additional cuts on contracted compound cut formulas, is avoided. To show termination of the reduction procedure an ordinal assignment based on techniques of Howard for Gödel’s T is used. (shrink)

We prove the first unconditional consistency result for superpolynomial circuit lower bounds with a relatively strong theory of bounded arithmetic. Namely, we show that the theory [Formula: see text] is consistent with the conjecture that [Formula: see text], i.e. some problem that is solvable in non-deterministic exponential time does not have polynomial size circuits. We suggest this is the best currently available evidence for the truth of the conjecture. The same techniques establish the same results with [Formula: see (...) text] replaced by the class of problems decidable in non-deterministic barely superpolynomial time such as [Formula: see text]. Additionally, we establish a magnification result on the hardness of proving circuit lower bounds. (shrink)

Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has been pressed in (...) the literature by Hilary Putnam and Peter Koellner. In this paper I sketch a conventionalist theory of mathematics, show that this conventionalist theory can meet the challenge just raised , and clarify the type of mathematical pluralism endorsed by the conventionalist by introducing the notion of a semantic counterpart. The paper’s aim is an improved understanding of conventionalism, pluralism, and the relationship between them. (shrink)