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Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization
 J. Symbolic Logic
, 1992
"... We show how to extract effective bounds Φ for ∀u1∀v ≤γ tu∃wηG0–sentences which depend on u only (i.e. ∀u∀v ≤γ tu∃w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form (∗)∀xδ∃y ≤ρ sx∀zτF0 (δ, ρ, τ are arbitrary finite types, η ≤ 2, G0, F0 are quantifier–free and s, t close ..."
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We show how to extract effective bounds Φ for ∀u1∀v ≤γ tu∃wηG0–sentences which depend on u only (i.e. ∀u∀v ≤γ tu∃w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form (∗)∀xδ∃y ≤ρ sx∀zτF0 (δ, ρ, τ are arbitrary finite types, η ≤ 2, G0, F0 are quantifier–free and s, t closed terms). If τ ≤ 2, (∗) can be weakened to ∀xδ, zτ∃y ≤ρ sx∀z ̃ ≤τ zF0. This is used to establish new conservation results about weak Knig’s lemma WKL. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers. 1 Introduction and basic notions Various theorems in classical analysis have the form A ≡ ∀x ∈ X∃y ∈ Kx ⊆ Y A1(x, y), where X,Y are complete separable metric spaces, Kx is compact in Y and A1 is purely universal. If an analytical sentence B is proved in using besides lemmata A only arithmetical constructions
A ModelTheoretic Approach to Ordinal Analysis
 Bulletin of Symbolic Logic
, 1997
"... . We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in no ..."
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. We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first and secondorder arithmetic. x1. Introduction. Two of proof theory's defining goals are the justification of classical theories on constructive grounds, and the extraction of constructive information from classical proofs. Since Gentzen, ordinal analysis has been a major component in these pursuits, and the assignment of recursive ordinals to theories has proven to be an illuminating way of measuring their constructive strength. The traditional approach to ordinal analysis, which uses cutelimination procedures to transfor...
From bounded arithmetic to second order arithmetic via automorphisms
 Logic in Tehran, Lect. Notes Log
"... Abstract. In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as P A, ACA0 (arithmetical comprehension schema with restricted induction), and Z2 (second order arithmetic). For example, we establish the following ..."
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Abstract. In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as P A, ACA0 (arithmetical comprehension schema with restricted induction), and Z2 (second order arithmetic). For example, we establish the following characterization of P A by proving a “reversal ” of a theorem of Gaifman: Theorem. The following are equivalent for completions T of I∆0: (a) T ⊢ P A; (b) Some model M = (M, · · ·) of T has a proper end extension N which satisfies I∆0 and for some automorphism j of N, M is precisely the fixed point set of j. Our results also shed light on the metamathematics of the QuineJensen system NF U of set theory with a universal set. 1.
Finite Models of Elementary Recursive Nonstandard Analysis
, 1996
"... This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in [2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Recursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERN ..."
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This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in [2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Recursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERNA proves the main axioms of the Chuaqui and Suppes system. Then a finitary consistency proof of ERNA will be given; in particular, we will show that PRA, the system of primitive recursive arithmetic, which is generally recognized as capturing Hilbert's notion of finitary, proves the consistency of ERNA. From the consistency proof we can extract a constructive method for obtaining finite approximations of models of nonstandard analysis. We present an isomorphism theorem for models that are finite substructures of infinite models. 1 Introduction This paper continues and extends the development of a constructive system of nonstandard analysis begun by Chuaqui and Suppes in [2, 9]. The ...
A standard model of Peano arithmetic with no conservative elementary extension, preprint
, 2006
"... Abstract. The principal result of this paper answers a longstanding question in the model theory of arithmetic [KS, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion ΩA: = (ω, +, ·, X)X∈A of the ..."
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Abstract. The principal result of this paper answers a longstanding question in the model theory of arithmetic [KS, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion ΩA: = (ω, +, ·, X)X∈A of the standard model of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension Ω ∗ A = (ω∗, · · ·) of ΩA, there is a subset of ω ∗ that is parametrically definable in Ω ∗ A but whose intersection with ω is not a member of A. Inspired by a recent question of Gitman and Hamkins, we also show that the aforementioned family A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra A/F IN (where F IN is the ideal of finite sets) collapses ℵ1 when viewed as a notion of forcing. 1.
A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' resul ..."
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Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a selfcontained proof requiring only basic knowledge of mathematical logic.
Generic cuts in models of arithmetic
 Mathematical Logic Quarterly
"... We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y. The notion of ‘indicator ’ is defined in a novel way, without initially specifying what property is indicated and is used to define a topological space of cu ..."
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We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y. The notion of ‘indicator ’ is defined in a novel way, without initially specifying what property is indicated and is used to define a topological space of cuts of the model. Various familiar properties of cuts (strength, regularity, saturation, coding properties) are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre. A new notion of ‘generic cut ’ is introduced and investigated and it is shown in the case of countable arithmetically saturated models M PA that generic cuts exist, indeed the set of generic cuts is comeagre in the sense of Baire, and furthermore that two generic cuts within the same ‘small interval ’ of the model are conjugate by an automorphism of the model. The paper concludes by outlining some applications to constructions of cuts satisfying properties incompatible with genericity, and discussing in modeltheoretic terms those properties for which there is an indicator Y. 1
Automorphisms of models of arithmetic: a unified view
 Ann. Pure Appl. Logic
"... 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 P A. In particular, we use this method to prove Theorem A below, which confirms a long standing conjectur ..."
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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 P A. 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 P A in which N is a strong cut, then for any M0 ≺ M there is an automorphism j of M such that the fixed point set of j is isomorphic to M0. We also finetune a number of classical results. One of our typical results in this direction is Theorem B below, which generalizes a theorem of KayeKossakKotlarski (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 P A in which N is a strong cut. There is a group embedding j ↦ → ˆj from
On a Question of Brown and Simpson
, 1994
"... this paper, we introduce an equivalent formulation of BCTII , which we denote BCT# ..."
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this paper, we introduce an equivalent formulation of BCTII , which we denote BCT#
The Termite and the Tower: Goodstein sequences and provability in PA
, 2007
"... We discuss Goodstein’s Theorem, a true finitary statement that can only be proven by infinitary means. We assume very little knowledge of logic and provide the necessary background to understand both Goodstein’s Theorem and, at a high level, how to show that a statement is true but not provable ins ..."
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We discuss Goodstein’s Theorem, a true finitary statement that can only be proven by infinitary means. We assume very little knowledge of logic and provide the necessary background to understand both Goodstein’s Theorem and, at a high level, how to show that a statement is true but not provable inside a given set of axioms. 1