Set theory determined by $V_\alpha$ for limit ordinals $\alpha>\omega$
Von Neumann hierarchy has a critical role in set theory. It is well-known that $V_\alpha$ is a model of $\mathsf{ZC}$ if $\alpha$ is a limit ordinal. Furthermore, $V_\alpha$ satisfies the cumulative...
View ArticleAnswer by Joel David Hamkins for Set theory determined by $V_\alpha$ for...
The sets $V_\lambda$ for limit ordinals $\lambda>\omega$ are precisely the models of second-order Zermelo set theory (including foundation scheme, infinity, and choice) plus the cumulative hierarchy...
View ArticleAnswer by Emil Jeřábek for Set theory determined by $V_\alpha$ for limit...
I don’t have a comprehensive answer, but a couple of observations:$T$ is included in ZFC unconditionally (i.e., not assuming the consistency of worldly cardinals).Indeed, for any sentence $\sigma$, ZFC...
View ArticleAnswer by Ali Enayat for Set theory determined by $V_\alpha$ for limit...
This answer concerns the second question (which asks whether there is a characterization of the theory of models of the form $V_{\alpha}$, where $\alpha$ ranges over limit ordinals), and its...
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