# Counting models of set theory

Fundamenta Mathematicae (2002)

• Volume: 174, Issue: 1, page 23-47
• ISSN: 0016-2736

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## Abstract

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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 $\mu \left(T,\alpha \right)={2}^{\aleph ₀}$. (3) If α < ω₁ and T ⊢ V ≠ OD, then $\mu \left(T,\alpha \right)\in 0,{2}^{\aleph ₀}$. (4) If τ is not well-ordered then $\mu \left(T,\tau \right)\in 0,{2}^{\aleph ₀}$. (5) If ZFC + “there is a measurable cardinal” has a well-founded model of height α < ω₁, then $\mu \left(T,\alpha \right)={2}^{\aleph ₀}$ for some complete extension T of ZF + V = OD.

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