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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 . There is some completion T of ZF such that μ(T) = κ.
(2) If α <ω₁ and μ(T,α) > ℵ₀, then .
(3) If α < ω₁ and T ⊢ V ≠ OD, then .
(4)...
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