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- Comment by Victoria Gitman on Consistency strength of the existence of a transitive model of $\mathsf{ZFC}^-$ with a $\kappa$-complete ultrafilterThis is really very nice!Victoria Gitman
- Comment by Victoria Gitman on Consistency strength of the existence of a transitive model of $\mathsf{ZFC}^-$ with a $\kappa$-complete ultrafilterAli, I am just seeing the comment because I have been away from MO for a while. So you are saying that even an ill-founded ultrapower by a non-normal filter as long as it is weakly amenable can be used to show that kappa is weakly compact in M?Victoria Gitman
- Answer by Victoria Gitman for Consistency strength of the existence of a transitive model of $\mathsf{ZFC}^-$ with a $\kappa$-complete ultrafilterLet's make some additional assumptions on the ultrafilter $U$. Suppose $M\models{\rm ZFC}^-$ and $\kappa$ is a cardinal in $M$. We say that $U$ is an $M$-ultrafilter if $\langle M,\in,U\rangle$ satisfies that $U$ is a $\kappa$-complete normal ultrafilter on $\kappa$. Because $U$ is only $\kappa$-complete for sequences in $M$ and $M$ might be missing even countable […]Victoria Gitman

- Comment by Victoria Gitman on Consistency strength of the existence of a transitive model of $\mathsf{ZFC}^-$ with a $\kappa$-complete ultrafilter
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# Monthly Archives: October 2013

## Forcing to add proper classes to a model of ${\rm GBC}$: The technicalities

In the previous post Forcing to add proper classes to a model of ${\rm GBC}$: An introduction, I made several sweeping assertions that will now be held up to public scrutiny.

## Forcing to add proper classes to a model of ${\rm GBC}$: An introduction

If you are interested in a mathematical universe whose ontology includes both sets and classes, you might consider for its foundation the ${\rm GBC}$ (Gödel-Bernays) axioms.