Regularity in models of arithmetic

1984 ◽  
Vol 49 (1) ◽  
pp. 272-280 ◽  
Author(s):  
George Mills ◽  
Jeff Paris
Keyword(s):  
Peano Arithmetic ◽  
Second Order ◽  
Axiom Schema ◽  
First Order ◽  
Nonstandard Models ◽  
Order Arithmetic ◽  
Bounded Sets ◽  
Models Of Arithmetic

AbstractThis paper investigates the quantifier “there exist unboundedly many” in the context of first-order arithmetic. An alternative axiomatization is found for Peano arithmetic based on an axiom schema of regularity: The union of boundedly many bounded sets is bounded. We also obtain combinatorial equivalents of certain second-order theories associated with cuts in nonstandard models of arithmetic.

Proof theory

2018 ◽  
Author(s):  
Wilfried Sieg
Keyword(s):  
Finite Type ◽  
Set Theory ◽  
Proof Theory ◽  
Second Order ◽  
Von Neumann ◽  
Second Order Arithmetic ◽  
First Order ◽  
David Hilbert ◽  
Heyting Arithmetic ◽  
Order Arithmetic

Proof theory is a branch of mathematical logic founded by David Hilbert around 1920 to pursue Hilbert’s programme. The problems addressed by the programme had already been formulated, in some sense, at the turn of the century, for example, in Hilbert’s famous address to the First International Congress of Mathematicians in Paris. They were closely connected to the set-theoretic foundations for analysis investigated by Cantor and Dedekind – in particular, to difficulties with the unrestricted notion of system or set; they were also related to the philosophical conflict with Kronecker on the very nature of mathematics. At that time, the central issue for Hilbert was the ‘consistency of sets’ in Cantor’s sense. Hilbert suggested that the existence of consistent sets, for example, the set of real numbers, could be secured by proving the consistency of a suitable, characterizing axiom system, but indicated only vaguely how to give such proofs model-theoretically. Four years later, Hilbert departed radically from these indications and proposed a novel way of attacking the consistency problem for theories. This approach required, first of all, a strict formalization of mathematics together with logic; then, the syntactic configurations of the joint formalism would be considered as mathematical objects; finally, mathematical arguments would be used to show that contradictory formulas cannot be derived by the logical rules. This two-pronged approach of developing substantial parts of mathematics in formal theories (set theory, second-order arithmetic, finite type theory and still others) and of proving their consistency (or the consistency of significant sub-theories) was sharpened in lectures beginning in 1917 and then pursued systematically in the 1920s by Hilbert and a group of collaborators including Paul Bernays, Wilhelm Ackermann and John von Neumann. In particular, the formalizability of analysis in a second-order theory was verified by Hilbert in those very early lectures. So it was possible to focus on the second prong, namely to establish the consistency of ‘arithmetic’ (second-order number theory and set theory) by elementary mathematical, ‘finitist’ means. This part of the task proved to be much more recalcitrant than expected, and only limited results were obtained. That the limitation was inevitable was explained in 1931 by Gödel’s theorems; indeed, they refuted the attempt to establish consistency on a finitist basis – as soon as it was realized that finitist considerations could be carried out in a small fragment of first-order arithmetic. This led to the formulation of a general reductive programme. Gentzen and Gödel made the first contributions to this programme by establishing the consistency of classical first-order arithmetic – Peano arithmetic (PA) – relative to intuitionistic arithmetic – Heyting arithmetic. In 1936 Gentzen proved the consistency of PA relative to a quantifier-free theory of arithmetic that included transfinite recursion up to the first epsilon number, ε0; in his 1941 Yale lectures, Gödel proved the consistency of the same theory relative to a theory of computable functionals of finite type. These two fundamental theorems turned out to be most important for subsequent proof-theoretic work. Currently it is known how to analyse, in Gentzen’s style, strong subsystems of second-order arithmetic and set theory. The first prong of proof-theoretic investigations, the actual formal development of parts of mathematics, has also been pursued – with a surprising result: the bulk of classical analysis can be developed in theories that are conservative over (fragments of) first-order arithmetic.

On generalized quantifiers in arithmetic

1982 ◽  
Vol 47 (1) ◽  
pp. 187-190 ◽  
Author(s):  
Carl Morgenstern
Keyword(s):  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
First Order ◽  
Order Language ◽  
Truth Definition ◽  
Order Arithmetic ◽  
Infinite Set

In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of L1, the language of first-order arithmetic, L, with the additional quantifer Ux which denotes “there are infinitely many x such that…. Note that Ux is the same thing as the Keisler quantifier Qx in the ℵ0 interpretation.We consider L2, which is L together with the ℵ0 interpretation of the Magidor-Malitz quantifier Q2xy which denotes “there is an infinite set X such that for distinct x, y ∈ X …”. In [1] Magidor and Malitz presented an axiom system for languages which arise from adding Q2 to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming ◊ω1, that the axioms are complete in the ℵ1 interpretation.If we let denote Peano arithmetic in L2 with induction for L2 formulas and the Magidor-Malitz axioms as logical axioms, we show that in we can give a truth definition for first-order Peano arithmetic, . Consequently we can prove in that is Πn sound for every n, thus in we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf.

NONSTANDARD MODELS IN RECURSION THEORY AND REVERSE MATHEMATICS

2014 ◽  
Vol 20 (2) ◽  
pp. 170-200 ◽  
Author(s):  
C. T. CHONG ◽  
WEI LI ◽  
YUE YANG
Keyword(s):  
Recursion Theory ◽  
Reverse Mathematics ◽  
Second Order ◽  
Ramsey's Theorem ◽  
Second Order Arithmetic ◽  
Ramsey’S Theorem ◽  
Nonstandard Models ◽  
Combinatorial Principles ◽  
Order Arithmetic ◽  
Effective Computability

AbstractWe give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

scholarly journals Interpreting the compositional truth predicate in models of arithmetic

2021 ◽  
Author(s):  
Cezary Cieśliński
Keyword(s):  
Proof Theory ◽  
Formal Proof ◽  
Truth Predicate ◽  
Saturated Model ◽  
First Order ◽  
Order Arithmetic ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Arithmetic

AbstractWe present a construction of a truth class (an interpretation of a compositional truth predicate) in an arbitrary countable recursively saturated model of first-order arithmetic. The construction is fully classical in that it employs nothing more than the classical techniques of formal proof theory.

scholarly journals Multi-sorted version of second order arithmetic

2016 ◽  
Vol 13 (5) ◽  
Author(s):  
Farida Kachapova
Keyword(s):  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Second Order ◽  
Natural Numbers ◽  
Natural Form ◽  
Second Order Arithmetic ◽  
Arithmetical Truth ◽  
Order Arithmetic

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.

A reflection principle and its applications to nonstandard models

1995 ◽  
Vol 60 (4) ◽  
pp. 1137-1152
Author(s):  
James H. Schmerl
Keyword(s):  
Number Theory ◽  
Reflection Principle ◽  
Peano Arithmetic ◽  
Second Order ◽  
Order Number ◽  
Nonstandard Models ◽  
Consistent Extension ◽  
Weak Theories

Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if T is any consistent extension of CA + AC and cf (λ) > ℵ0, then T has a natural λ-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of . (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11].A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.

Forcing and reducibilities. III. Forcing in fragments of set theory

1983 ◽  
Vol 48 (4) ◽  
pp. 1013-1034
Author(s):  
Piergiorgio Odifreddi
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
Second Order ◽  
Analytic Hierarchy ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Language ◽  
Order Arithmetic ◽  
Definition Of ◽  
Generic Set

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:A0 = {X ⊆ ω: X is arithmetic},Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},Aλ = ⋃α<λAα (λ limit),RA = ⋃αAα.We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:L0 = ⊘,Lα+1 = {X: X is (first-order) definable over Lα},Lλ = ⋃α<λLα (λ limit),L = ⋃αLα.

Ordinal bounds for κ-consistency

1974 ◽  
Vol 39 (4) ◽  
pp. 693-699 ◽  
Author(s):  
Warren D. Goldfarb
Keyword(s):  
Nonnegative Integer ◽  
Transfinite Induction ◽  
Axiom Schema ◽  
First Order ◽  
Order Arithmetic ◽  
Definition Of ◽  
Image Position ◽  
Primitive Recursive

In [1] the ω-consistency of arithmetic was proved by a method which yields fine ordinal bounds for κ-consistency, κ ≥ 1. In this paper these bounds are shown to be best possible. The ω-consistency of a number-theoretic system S can be expressed thus: for all sentences ∃xM,where ProvS is the proof predicate for S, if n is a nonnegative integer then n is the formal numeral (of S) for n, and if G is a formula then ˹G˺ is the Gödel number of G. The κ-consistency of S is the restriction of (1) to Σκ0 sentences ∃xM. The proof in [1] establishes the no-counterexample interpretation of (1), that is, the existence of a constructive functional Φ such that, for all sentences ∃xM, all numbers p, and all functions f,(see [1, §2]). A functional Φ is an ω-consistency functional for S if it satisfies (2) for all sentences ∃xM, and a κ-consistency functional for S if it satisfies (2) for all Σκ0 sentences ∃xM.The systems considered in [1] are those obtained from classical first-order arithmetic Z, including the schema for definition of primitive recursive (p.r.) functions, by adjoining, for some p.r. well-founded ordering ≺ of the nonnegative integers, the axiom schemathat is, the least number principle on ≺; it is equivalent to the schema of transfinite induction on ≺.

James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - James H. Schmerl. Automorphism groups of models of Peano arithmetic. The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1249–1264. - James H. Schmerl. Diversity in substructures. Nonstandard models of arithmetic and set theory. edited by A. Enayat and R. Kossak, Contemporary Mathematics, vol. 361, American Mathematical Societey (2004), pp. 45–161. - James H. Schmerl. Generic automorphisms and graph coloring. Discrete Mathematics, vol. 291 (2005), no. 1–3, pp. 235–242. - James H. Schmerl. Nondiversity in substructures. The Journal of Symbolic Logic, vol. 73 (2008), no. 1, pp. 193–211.

2009 ◽  
Vol 15 (2) ◽  
pp. 222-227
Author(s):  
Roman Kossak
Keyword(s):  
Set Theory ◽  
Graph Coloring ◽  
Automorphism Groups ◽  
Peano Arithmetic ◽  
Discrete Mathematics ◽  
Symbolic Logic ◽  
Nonstandard Models ◽  
Recursively Saturated ◽  
University Of Connecticut ◽  
Models Of Arithmetic

An infinitistic rule of proof

1968 ◽  
Vol 32 (4) ◽  
pp. 447-451 ◽  
Author(s):  
H. B. Enderton
Keyword(s):  
Formal System ◽  
Peano Arithmetic ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic

In this paper we consider a fonnal system of second-order Peano arithmetic with a rule of inference stronger than the ω-rule [3]. We also consider the relation to a class of models for analysis (i.e. second-order arithmetic) which lies between the class of ω-models and the class of β-models [5].The notation used is largely that of [3] and [5]. We assume that the reader has some familiarity with at least the ideas of the former. The formal system (A) of Peano arithmetic employed in [3] includes the comprehension axioms and the second-order induction axiom.

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