Counting models of set theory
Volume 174 / 2002
Fundamenta Mathematicae 174 (2002), 23-47
MSC: Primary 03C62, 03C15; Secondary 03H99.
DOI: 10.4064/fm174-1-2
Abstract
Let $T$ denote a completion of ZF. We are interested in the number $\mu (T) $ of isomorphism types of countable well-founded models of $T$. Given any countable order type $\tau $, we are also interested in the number $\mu (T,\tau )$ of isomorphism types of countable models of $T$ whose ordinals have order type $\tau $. 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 $\mu (T)=\kappa$.
$(2)$ If $\alpha <\omega _{1}$ and $\mu (T,\alpha )>\aleph _{0}$, then $\mu (T,\alpha )=2^{\aleph _{0}}$.$(3)$ If $\alpha <\omega _{1}$ and $T\vdash {\bf V} \neq {\bf OD}$, then $\mu (T,\alpha )\in \{0,2^{\aleph _{0}}\}$.
$(4)$ If $\tau $ is not well-ordered then $\mu (T,\tau )\in \{0,2^{\aleph _{0}}\}$.
$(5)$ If ZFC $+$ “there is a measurable cardinal” has a well-founded model of height $\alpha <\omega _{1}$, then $\mu (T,\alpha )=2^{\aleph _{0}}$ for some complete extension $T$ of $\hbox{ZF}+\mathbf{V}=\mathbf{OD}$.
