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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 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' result is ..."
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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.
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#
Ordertypes of models of Peano arithmetic: a short survey
, 2001
"... This paper is a short and slightly selective survey of results on ordertypes of models of Peano arithmetic. We include few proofs, and concentrate instead on the key problems as we see them and possible ways of responding to the very considerable mathematical di#culties raised ..."
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This paper is a short and slightly selective survey of results on ordertypes of models of Peano arithmetic. We include few proofs, and concentrate instead on the key problems as we see them and possible ways of responding to the very considerable mathematical di#culties raised
Σ_nBounding and Δ_nInduction
 Proc. Amer. Math. Soc
"... Working in the base theory of PA  exp, we show that for all n #, the bounding principle for # n formulas (B# n ) is equivalent to the induction principle for # n formulas (I# n ). This partially answers a question of J. Paris; see Clote and Kraj cek (1993). ..."
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Working in the base theory of PA  exp, we show that for all n #, the bounding principle for # n formulas (B# n ) is equivalent to the induction principle for # n formulas (I# n ). This partially answers a question of J. Paris; see Clote and Kraj cek (1993).
FACTORIZATION OF POLYNOMIALS AND ~1 INDUCTION*
, 1985
"... In the body of this paper we use the apparatus of mathematical logic to investigate the role of induction in algebraic reasoning. We show that a surprisingly strong form of induction is needed in order to prove certain very basic and simple algebraic lemmas. ..."
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In the body of this paper we use the apparatus of mathematical logic to investigate the role of induction in algebraic reasoning. We show that a surprisingly strong form of induction is needed in order to prove certain very basic and simple algebraic lemmas.