# Automorphisms of models of bounded arithmetic

Fundamenta Mathematicae (2006)

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

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topAli 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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