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• # Artykuł - szczegóły

## Fundamenta Mathematicae

2002 | 174 | 1 | 23-47

## Counting models of set theory

EN

### Abstrakty

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.

23-47

wydano
2002

### Twórcy

autor
• Department of Mathematics and Statistics, American University, Washington, DC 20016-8050, U.S.A.