# Counting models of set theory

Fundamenta Mathematicae (2002)

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

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topAli Enayat. "Counting models of set theory." Fundamenta Mathematicae 174.1 (2002): 23-47. <http://eudml.org/doc/282912>.

@article{AliEnayat2002,

abstract = {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.},

author = {Ali Enayat},

journal = {Fundamenta Mathematicae},

keywords = {isomorphism types of countable well-founded models of set theory},

language = {eng},

number = {1},

pages = {23-47},

title = {Counting models of set theory},

url = {http://eudml.org/doc/282912},

volume = {174},

year = {2002},

}

TY - JOUR

AU - Ali Enayat

TI - Counting models of set theory

JO - Fundamenta Mathematicae

PY - 2002

VL - 174

IS - 1

SP - 23

EP - 47

AB - 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.

LA - eng

KW - isomorphism types of countable well-founded models of set theory

UR - http://eudml.org/doc/282912

ER -

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