# Automorphisms of models of bounded arithmetic

Fundamenta Mathematicae (2006)

• Volume: 192, Issue: 1, page 37-65
• ISSN: 0016-2736

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## Abstract

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We establish the following model-theoretic characterization of the fragment IΔ₀ + Exp + BΣ₁ of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment IΔ₀ of Peano arithmetic with induction limited to Δ₀-formulae). Theorem A. The following two conditions are equivalent for a countable model of the language of arithmetic: (a) satisfies IΔ₀ + BΣ₁ + Exp; (b) $={I}_{fix}\left(j\right)$ for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here ${I}_{fix}\left(j\right)$ 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 Σ₁-collection 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 Δ₀-formula. 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 is a countable recursively saturated model of PA and I is a proper initial segment of that is closed under exponentiation. There is a group embedding j ↦ ĵ from Aut(ℚ) into Aut( ) such that $I={I}_{fix}\left(ĵ\right)$ for every nontrivial j ∈ Aut(ℚ). Moreover, if j is fixed point free, then the fixed point set of ĵ is isomorphic to . Here Aut(X) is the group of automorphisms of the structure X, and ℚ is the ordered set of rationals.

## How to cite

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Ali Enayat. "Automorphisms of models of bounded arithmetic." Fundamenta Mathematicae 192.1 (2006): 37-65. <http://eudml.org/doc/282846>.

@article{AliEnayat2006,
abstract = {We establish the following model-theoretic characterization of the fragment IΔ₀ + Exp + BΣ₁ of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment IΔ₀ of Peano arithmetic with induction limited to Δ₀-formulae). Theorem A. The following two conditions are equivalent for a countable model of the language of arithmetic: (a) satisfies IΔ₀ + BΣ₁ + Exp; (b) $= I_\{fix\}(j)$ for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here $I_\{fix\}(j)$ 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 Σ₁-collection 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 Δ₀-formula. 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 is a countable recursively saturated model of PA and I is a proper initial segment of that is closed under exponentiation. There is a group embedding j ↦ ĵ from Aut(ℚ) into Aut( ) such that $I = I_\{fix\}(ĵ)$ for every nontrivial j ∈ Aut(ℚ). Moreover, if j is fixed point free, then the fixed point set of ĵ is isomorphic to . Here Aut(X) is the group of automorphisms of the structure X, and ℚ is the ordered set of rationals.},
author = {Ali Enayat},
journal = {Fundamenta Mathematicae},
keywords = {first-order arithmetic; bounded arithmetic; recursive saturation; automorphism},
language = {eng},
number = {1},
pages = {37-65},
title = {Automorphisms of models of bounded arithmetic},
url = {http://eudml.org/doc/282846},
volume = {192},
year = {2006},
}

TY - JOUR
AU - Ali Enayat
TI - Automorphisms of models of bounded arithmetic
JO - Fundamenta Mathematicae
PY - 2006
VL - 192
IS - 1
SP - 37
EP - 65
AB - We establish the following model-theoretic characterization of the fragment IΔ₀ + Exp + BΣ₁ of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment IΔ₀ of Peano arithmetic with induction limited to Δ₀-formulae). Theorem A. The following two conditions are equivalent for a countable model of the language of arithmetic: (a) satisfies IΔ₀ + BΣ₁ + Exp; (b) $= I_{fix}(j)$ for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here $I_{fix}(j)$ 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 Σ₁-collection 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 Δ₀-formula. 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 is a countable recursively saturated model of PA and I is a proper initial segment of that is closed under exponentiation. There is a group embedding j ↦ ĵ from Aut(ℚ) into Aut( ) such that $I = I_{fix}(ĵ)$ for every nontrivial j ∈ Aut(ℚ). Moreover, if j is fixed point free, then the fixed point set of ĵ is isomorphic to . Here Aut(X) is the group of automorphisms of the structure X, and ℚ is the ordered set of rationals.
LA - eng
KW - first-order arithmetic; bounded arithmetic; recursive saturation; automorphism
UR - http://eudml.org/doc/282846
ER -

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