Counting models of set theory

Ali Enayat

Fundamenta Mathematicae, Tome 173 (2002), p. 23-47 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2002-01-01

Zbl 0998.03033

EUDML-ID : urn:eudml:doc:282912