Arithmetic PA Research Articles

Serial properties, selector proofs and the provability of consistency

Abstract The consistency of a theory means that each of its formal derivations $D_{0}, D_{1}, D_{2}, \ldots $ is free of contradictions. For Peano Arithmetic PA, after the standard coding of derivations by numerals, PA-consistency is directly represented by the consistency scheme $\textsf{Con}^{S}_{\textsf{PA}}$, which is a series of arithmetical statements ‘$n$ is not a code of a derivation of $\ (0=1)$’ for numerals $n=0,1,2,\ldots $. We note that the consistency formula $\textsf{Con}_{\textsf{PA}}$, $\forall x$ ‘$x$ is not a code of a derivation of $(0=1)$,’ is strictly stronger in PA than PA-consistency and corresponds to some other property, which we call uniform consistency. When studying the provability of consistency in PA we ought to work not with the consistency formula $\textsf{Con}_{\textsf{PA}}$ but rather with the consistency scheme $\textsf{Con}^{S}_{\textsf{PA}}$, which adequately represents PA-consistency. This paper introduces the Hilbert-inspired notion of proof of an infinite series of formulas in a theory and proves PA-consistency in the form $\textsf{Con}^{S}_{\textsf{PA}}$ in PA. These findings show that PA proves its consistency whereas, by Gödel’s second incompleteness theorem, PA cannot prove its uniform consistency.

Every Countable Model of Arithmetic or Set Theory has a Pointwise-Definable End Extension

Abstract According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its Σ n generalizations to build a progressively elementary tower making any desired individual a n definable at each stage n, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve V = L in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of ZFC + V = L[μ].

A single set improvement to the 3k − 4 theorem

The 3k−4 Theorem is a classical result which asserts that if A,B⊆Z are finite, nonempty subsets with(1)|A+B|=|A|+|B|+r≤|A|+|B|+min⁡{|A|,|B|}−3−δ, where δ=1 if A and B are translates of each other, and otherwise δ=0, then there are arithmetic progressions PA and PB of common difference such that A⊆PA, B⊆PB, |B|≤|PB|+r+1 and |PA|≤|A|+r+1. It is one of the few cases in Freiman's Theorem for which exact bounds on the sizes of the progressions are known. The hypothesis (1) is best possible in the sense that there are examples of sumsets A+B having cardinality just one more than that of (1), yet A and B cannot both be contained in short length arithmetic progressions. In this paper, we show that the hypothesis (1) can be significantly weakened and still yield the same conclusion for one of the sets A and B. Specifically, if |B|≥3, s≥1 is the unique integer with(s−1)s(|B|2−1)+s−1<|A|≤s(s+1)(|B|2−1)+s, and(2)|A+B|=|A|+|B|+r<(|A|s+|B|2−1)(s+1), then we show there is an arithmetic progression PB⊆Z with B⊆PB and |PB|≤|B|+r+1. The hypothesis (2) is best possible (without additional assumptions on A) for obtaining such a conclusion.

Axiomatizing Provable n-Provability

The set of all formulas whose n-provability in a given arithmetical theory S is provable in another arithmetical theory T is a recursively enumerable extension of S. We prove that such extensions can be naturally axiomatized in terms of transfinite progressions of iterated local reflection schemata over S. Specifically, the set of all provably 1-provable sentences in Peano arithmetic PA can be axiomatized by an e0-times iterated local reflection schema over PA. The resulting characterizations provide additional information on the proof-theoretic strength of these theories and on the complexity of their axiomatization.

The truth assignments that differentiate human reasoning from mechanistic reasoning: The evidence-based argument for Lucas’ Gödelian thesis

We consider the argument that Tarski’s classic definitions permit an intelligence—whether human or mechanistic—to admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1) in terms of classical algorithmic verifiabilty; and (2) in terms of finitary algorithmic computability. We then show that the two definitions correspond to two distinctly different assignments of satisfaction and truth to the compound formulas of PA over N—IPA(N,SV) and IPA(N,SC). We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both IPA(N,SV) and IPA(N,SC). We then show: (a) that if we assume the satisfaction and truth of the compound formulas of PA are always non-finitarily decidable under IPA(N,SV), then this assignment corresponds to the classical non-finitary standard interpretation IPA(N,S) of PA over the domain N; and (b) that the satisfaction and truth of the compound formulas of PA are always finitarily decidable under the assignment IPA(N,SC), from which we may finitarily conclude that PA is consistent. We further conclude that the appropriate inference to be drawn from Gödel’s 1931 paper on undecidable arithmetical propositions is that we can define PA formulas which—under any interpretation over N—are algorithmically verifiable as always true over N, but not algorithmically computable as always true over N. We conclude from this that Lucas’ Gödelian argument is validated if the assignment IPA(N,SV) can be treated as circumscribing the ambit of human reasoning about ‘true’ arithmetical propositions, and the assignment IPA(N,SC) as circumscribing the ambit of mechanistic reasoning about ‘true’ arithmetical propositions.

Interpretation of constructive multi-typed theory in the theory of arithmetical truth

This paper considers an axiomatic theory BT, which can be used to formalise constructive mathematics. BT has an intuitionistic logic, combinatorial operations and sets of many types. BT has such constructive features as a predicative comprehension axiom and consistency with the formal Church thesis but BT is also consistent with classical logic. In addition to the properties of BT studied before, in this paper we study the proof-theoretical strength of BT and its fragments by constructing an interpretation of BT in a so called theory of arithmetical truth PATr, which is obtained from the Peano arithmetic PA by adding infinitely many truth predicates.

Completeness of Hoare logic with inputs over the standard model

Hoare logic for the set of while-programs with the first-order logical language L and the first-order theory T⊂L is denoted by HL(T). Bergstra and Tucker have pointed out that the complete number theory Th(N) is the only extension T of Peano arithmetic PA for which HL(T) is logically complete. The completeness result is not satisfying, since it allows inputs to range over nonstandard models. The aim of this paper is to investigate under what circumstances HL(T) is logically complete when inputs range over the standard model N. PA+ is defined by adding to PA all the unprovable Π1-sentences that describe the nonterminating computations. It is shown that each computable function in N is uniformly Σ1-definable in all models of PA+, and that PA+ is arithmetical. Finally, it is established, based on the reduction from HL(T) to T, that PA+ is the minimal extension T of PA for which HL(T) is logically complete when inputs range over N. This completeness result has an advantage over Bergstra's and Tucker's one, in that PA+ is arithmetical while Th(N) is not.

On elementary theories of ordinal notation systems based on reflection principles

L.D. Beklemishev has recently introduced a constructive ordinal notation system for the ordinal ɛ 0. We consider this system and its fragments for smaller ordinals ω n (towers of ω-exponentiations of height n). These systems are based on Japaridze’s well-known polymodal provability logic. They are used in the technique of ordinal analysis of the Peano arithmetic PA and its fragments on the basis of iterated reflection schemes. Ordinal notation systems can be regarded as models of the first-order language. We prove that the full notation system and its fragments for ordinals ≥ω 4 have undecidable elementary theories. At the same time, the fragments of the full system for ordinals ≤ω 3 have decidable elementary theories. We also obtain results on decidability of the elementary theory for ordinal notation systems with weaker signatures.

Proof theory and mathematical meaning of paraconsistent C-systems

A proof-theoretic analysis and new arithmetical semantics are proposed for some paraconsistent C-systems, which are a relevant sub-class of Logics of Formal Inconsistency ( LFIs) introduced by W.A. Carnielli et al. (2002, 2005) [8,9]. The sequent versions BC, CI, CIL of the systems bC, Ci, Cil presented in Carnielli et al. (2002, 2005) [8,9] are introduced and examined. BC, CI, CIL admit the cut-elimination property and, in general, a weakened sub-formula property. Moreover, a formal notion of constructive paraconsistent system is given, and the constructivity of CI is proven. Further possible developments of proof theory and provability logic of CI-based arithmetical systems are sketched, and a possible weakened Hilbertʼs program is discussed. As to the semantical aspects, arithmetical semantics interprets C-system formulas into Provability Logic sentences of classical Arithmetic PA (Artemov and Beklemishev (2004) [2], Japaridze and de Jongh (1998) [19], Gentilini (1999) [15], Smorynski (1991) [22]): thus, it links the notion of truth to the notion of provability inside a classical environment. It makes true infinitely many contradictions B ∧ ¬ B and falsifies many arbitrarily complex instances of non-contradiction principle ¬ ( A ∧ ¬ A ) . Moreover, arithmetical models falsify both classical logic LK and intuitionistic logic LJ, so that a kind of metalogical completeness property of LFI-paraconsistent logic w.r.t. arithmetical semantics is proven. As a work in progress, the possibility to interpret CI-based paraconsistent Arithmetic PACI into Provability Logic of classical Arithmetic PA is discussed, showing the role that PACI arithmetical models could have in establishing new meta-mathematical properties, e.g. in breaking classical equivalences between consistency statements and reflection principles.

Around provability logic

We present some results on algebraic and modal analysis of polynomial (intrinsically definable) distortions of the standard provability predicate in Peano Arithmetic PA, and investigate three provability-like modal systems related to the Gödel–Löb modal system GL. We also present a short review of relational and topological semantics for these systems, and describe the dual category of algebraic models of our main modal system.

Nonstandard models that are definable in models of Peano Arithmetic

AbstractIn this paper, we investigate definable models of Peano Arithmetic PA in a model of PA. For any definable model N without parameters in a model M, we show that N is isomorphic to M if M is elementary extension of the standard model and N is elementarily equivalent to M. On the other hand, we show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M but N is not isomorphic to M. We also show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M, and N is isomorphic to M, but N is not definably isomorphic to M. And also, we give a generalization of Tennenbaum's theorem. At the end, we give a new method to construct a definable model by a refinement of Kotlarski's method. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Automorphisms of models of arithmetic: A unified view

We develop the method of iterated ultrapower representation to provide a unified and perspicuous approach for building automorphisms of countable recursively saturated models of Peano arithmetic PA . In particular, we use this method to prove Theorem A below, which confirms a long-standing conjecture of James Schmerl. Theorem A If M is a countable recursively saturated model of PA in which N is a strong cut, then for any M 0 ≺ M there is an automorphism j of M such that the fixed point set of j is isomorphic to M 0 . We also fine-tune a number of classical results. One of our typical results in this direction is Theorem B below, which generalizes a theorem of Kaye–Kossak–Kotlarski (in what follows Aut ( X ) is the automorphism group of the structure X , and Q is the ordered set of rationals). Theorem B Suppose M is a countable recursively saturated model of PA in which N is a strong cut. There is a group embedding j ↦ j ˆ from Aut ( Q ) into Aut ( M ) such that for each j ∈ Aut ( Q ) that is fixed point free, j ˆ moves every undefinable element of M .

Numbers and Polynomials- 50 years since the publication of Wittgenstein's Bemerkungen über die Grundlagen der Mathematik (1956): Mathematical and Educational reflections

According to L. Wittgenstein, the meaning of a mathematical object is to be grounded upon its use. In this paper we consider Robinson theory Q, the subtheory of firstorder Peano Arithmetic PA; some theorems and conjectures can be interpreted over one model of Q given by a universe of polynomials; with respect to nonconstant polynomials some proofs by elementary methods are given and compared with corresponding results in the standard model of PA. We conclude that the creative power of the language can be pointed out in how the language itself is embedded into the rest of human activities, and this is an important track to follow for researchers in mathematics education.

On uniform weak König's lemma

The so-called weak König's lemma WKL asserts the existence of an infinite path b in any infinite binary tree (given by a representing function f). Based on this principle one can formulate subsystems of higher-order arithmetic which allow to carry out very substantial parts of classical mathematics but are Π 2 0 -conservative over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In Kohlenbach [10] (J. Symbolic Logic 57 (1992) 1239–1273) we established such conservation results relative to finite type extensions PRA ω of PRA (together with a quantifier-free axiom of choice schema which—relative to PRA ω —implies the schema of Σ 1 0 -induction). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional Φ which selects uniformly in a given infinite binary tree f an infinite path Φf of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in Kohlenbach [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA ω only has a quantifier-free rule of extensionality QF-ER instead of the full axioms (E) of extensionality for all finite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be Π 2 0 -conservative over PRA, PRA ω+(E)+ UWKL contains (and is conservative over) full Peano arithmetic PA. We also investigate the proof–theoretic as well as the computational strength of UWKL relative to the intuitionistic variant of PRA ω both with and without the Markov principle.

Two Proof-Theoretic Remarks on EA + ECT

In this note two propositions about the epistemic formalization of Church's Thesis (ECT) are proved. First it is shown that all arithmetical sentences deducible in Shapiro's system EA of Epistemic Arithmetic from ECT are derivable from Peano Arithmetic PA + uniform reflection for PA. Second it is shown that the system EA + ECT has the epistemic disjunction property and the epistemic numerical existence property for arithmetical formulas.

The liar paradox and fuzzy logic

AbstractCan one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying “xis true” and satisfying the “dequotation schema”for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.

On the No-Counterexample Interpretation

AbstractIn [15], [16] G. Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated ε-substitution method (due to W. Ackermann), that for every theoremA(Aprenex) of first-order Peano arithmeticPAone can find ordinal recursive functionalsof order type &lt; ε0which realize the Herbrand normal formAHofA.Subsequently more perspicuous proofs of this fact via functional interpretation (combined with normalization) and cut-elimination were found. These proofs however do not carry out the no-counterexample interpretation as alocalproof interpretation and don't respect the modus ponens on the level of the nocounterexample interpretation of formulasAandA → B. Closely related to this phenomenon is the fact that both proofs do not establish the condition (δ) and—at least not constructively—(γ) which are part of the definition of an ‘interpretation of a formal system’ as formulated in [15].

On the Uniform Weak König’s Lemma

The so-called weak K¨onig's lemma WKL asserts the existence of an infinite&lt;br /&gt;path b in any infinite binary tree (given by a representing function f). Based on&lt;br /&gt;this principle one can formulate subsystems of higher-order arithmetic which&lt;br /&gt;allow to carry out very substantial parts of classical mathematics but are PI^0_2-conservative&lt;br /&gt;over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In [10] we established such conservation results relative to finite type extensions PRA^omega of PRA (together with a quantifier-free axiom of choice schema). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional Phi which selects uniformly in a given infinite binary tree f an infinite path Phi f of that tree.&lt;br /&gt;This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in [10] actually can be used to eliminate even this uniform weak K¨onig's lemma provided that PRA^omega only has a quantifier-free rule of extensionality QF-ER instead of the full axioms (E) of extensionality for all finite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be Pi^0_2 -conservative over PRA, PRA^omega +(E)+UWKL contains (and is conservative over) full Peano arithmetic PA.

Systems of explicit mathematics with non-constructive μ-operator. Part II

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET(μ) plus set induction (type induction, formula induction) is proof-theoretically equivalent to Peano arithmetic PA (the second-order system ( Π 0 ∞-CA < ε 0 , the second-order system ( Π 0 ∞-CA) < ε ε 0 ).

Totality in applicative theories

In this paper we study applicative theories of operations and numbers with (and without) the non-constructive minimum operator in the context of a total application operation. We determine the proof-theoretic strength of such theories by relating them to well-known systems like Peano Arithmetic PA and the system ( Π ∞ 0- CA) < ε 0 of second order arithmetic. Essential use will be made of so-called fixed-point theories with ordinals, certain infinitary term models and Church-Rosser properties.