Counting models of set theory

Ali Enayat¹

¹ Department of Mathematics and Statistics American University Washington, DC 20016-8050, U.S.A.

Fundamenta Mathematicae, Tome 174 (2002) no. 1, pp. 23-47

Résumé

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 $\hbox{ZF}+\mathbf{V}=\mathbf{OD}$.

Détail

DOI : 10.4064/fm174-1-2

Keywords: denote completion interested number isomorphism types countable well founded models given countable order type tau interested number tau isomorphism types countable models whose ordinals have order type tau prove suppose zfc has uncountable well founded model kappa omega cup aleph aleph aleph there completion kappa alpha omega alpha aleph alpha aleph alpha omega vdash neq alpha aleph tau well ordered tau aleph zfc there measurable cardinal has well founded model height alpha omega alpha aleph complete extension hbox mathbf mathbf