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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 $\kappa \in \omega \cup \aleph ₀,\aleph ₁,2^{\aleph ₀}$. There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $\mu (T,\alpha )=2^{\aleph ₀}$. (3) If α < ω₁ and T ⊢ V ≠ OD, then $\mu (T,\alpha )\in 0,2^{\aleph ₀}$. (4)...

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...

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 $(V_{\alpha },\in )$ 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)...