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Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Cited by 20 (6 self)
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Algebraic proofs of cut elimination
 Journal of Logic Algebraic Programming
"... Algebraic proofs of the cutelimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the doublenegation translation is also discussed: if ϕ ..."
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Cited by 12 (9 self)
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Algebraic proofs of the cutelimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the doublenegation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ) nf is provable in minimal logic, where θ nf denotes the negationnormal form of θ. The translation is used to show that cutelimination theorems for classical logic can be viewed as special cases of the cutelimination theorems for intuitionistic logic. 1
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
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Cited by 7 (0 self)
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
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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Cited by 3 (2 self)
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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.
Automorphisms of Models of Bounded Arithmetic ∗
, 2006
"... We establish the following model theoretic characterization of the fragment I∆0 +Exp+BΣ1 of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment I∆0 of Peano arithmetic with induction limited to ∆0formulae). Theorem A. The following two conditions ..."
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We establish the following model theoretic characterization of the fragment I∆0 +Exp+BΣ1 of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment I∆0 of Peano arithmetic with induction limited to ∆0formulae). Theorem A. The following two conditions are equivalent for a countable model M of the language of arithmetic: (a) M satisfies I∆0 + BΣ1 + Exp. (b) M = Ifix(j) for some nontrivial automorphism j of an end extension N of M that satisfies I∆0. Here Ifix(j) is the largest initial segment of the domain of j that is pointwise fixed by j, Exp is the axiom asserting the totality of the exponential function, and BΣ1 is the Σ1collection scheme consisting of the universal closure of formulae of the form [∀x < a ∃y ϕ(x, y)] → [∃z ∀x < a ∃y < z ϕ(x, y)], where ϕ is a ∆0formula. Theorem A was inspired by a theorem of Smoryński, but the method of proof of Theorem A is quite different and yields the following strengthening of Smoryński’s result: Theorem B. Suppose M is a countable recursively saturated model of P A and I is a proper initial segment of M that is closed under exponentiation. There is a group embedding j ↦− → ̂j from Aut(Q) into
“PARAMETER FREE Π1–INDUCTION AND RESTRICTED EXPONENTIATION” AND “A MODEL THEORETIC APPROACH TO PARAMETER FREE Π2–INDUCTION” (EXTENDED ABSTRACT)
, 2009
"... Abstract. This work falls into two parts. In the first part, we characterize the sets of all Π2 and all B(Σ1) ( = boolean combinations of Σ1) theorems of IΠ − 1 in terms of restricted exponentiation; and use these characterizations to prove that both sets are not deductively equivalent. In the secon ..."
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Abstract. This work falls into two parts. In the first part, we characterize the sets of all Π2 and all B(Σ1) ( = boolean combinations of Σ1) theorems of IΠ − 1 in terms of restricted exponentiation; and use these characterizations to prove that both sets are not deductively equivalent. In the second part, we investigate how these results generalize to IΠ − 2. We present a new model–theoretic approach to IΠ−2 that allows us to obtain informative characterizations of the set of all Π2 and all B(Σ2) theorems of IΠ − 2; and to give a model–theoretic proof of a conservation result by L. Beklemishev’s stating that IΠ − 2 is a B(Σ2) conservative extension of IΣ−1. 1.
Published In Algebraic proofs of cut elimination∗
, 2001
"... Algebraic proofs of the cutelimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the doublenegation translation is also discussed: if ..."
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Algebraic proofs of the cutelimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the doublenegation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negationnormal form of θ. The translation is used to show that cutelimination theorems for classical logic can be viewed as special cases of the cutelimination theorems for intuitionistic logic. 1