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1

Automorphisms of models of bounded arithmetic

100%

Enayat A.

Fundamenta Mathematicae

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2006

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tom 192

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nr 1

37-65

EN

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.

2

On the Leibniz-Mycielski axiom in set theory

100%

Enayat A.

Fundamenta Mathematicae

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2004

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tom 181

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nr 3

215-231

EN

Motivated by Leibniz's thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that $(V_{α},∈)$ satisfies φ(x) ∧¬ φ(y). We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows: 1. In the presence of ZF, the following are equivalent: (a) LM. (b) The existence of a parameter free definable class function F such that for all sets x with at least two elements, ∅ ≠ F(x) ⊊ x. (c) The existence of a parameter free definable injection of the universe into the class of subsets of ordinals. 2. Con(ZF) ⇒ Con(ZFC +¬LM). 3. [Solovay] Con(ZF) ⇒ Con(ZF + LM + ¬AC).

3

Counting models of set theory

100%

Enayat A.

Fundamenta Mathematicae

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2002

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tom 174

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nr 1

23-47

EN

Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and $κ ∈ ω ∪ {ℵ₀,ℵ₁,2^{ℵ₀}}$. There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $μ(T,α) = 2^{ℵ₀}$. (3) If α < ω₁ and T ⊢ V ≠ OD, then $μ(T,α) ∈ {0,2^{ℵ₀}}$. (4) If τ is not well-ordered then $μ(T,τ) ∈ {0,2^{ℵ₀}}$. (5) If ZFC + "there is a measurable cardinal" has a well-founded model of height α < ω₁, then $μ(T,α) = 2^{ℵ₀}$ for some complete extension T of ZF + V = OD.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Enayat A.

1 2006

1 2004

1 2002

Automorphisms of models of bounded arithmetic

On the Leibniz-Mycielski axiom in set theory

Counting models of set theory