# 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.

## How to cite

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Ali 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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