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2006 | 192 | 1 | 37-65
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Automorphisms of models of bounded arithmetic

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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 𝔐 of the language of arithmetic:
(a) 𝔐 satisfies IΔ₀ + BΣ₁ + Exp;
(b) $𝔐 = I_{fix}(j)$ for some nontrivial automorphism j of an end extension 𝔑 of 𝔐 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 𝔐 is a countable recursively saturated model of PA and I is a proper initial segment of 𝔐 that is closed under exponentiation. There is a group embedding j ↦ ĵ from Aut(ℚ) into Aut(𝔐 ) 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 𝔐.
Here Aut(X) is the group of automorphisms of the structure X, and ℚ is the ordered set of rationals.
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  • Department of Mathematics and Statistics, American University, 4400 Mass. Ave. N.W., Washington, DC 20016-8050, U.S.A.
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