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### Mathoverflow Activity

- 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 ultrafilterDo you require that the ultrapower is well-founded? Also, do you want $M$ to satisfy that the ultrafilter is normal?Victoria Gitman

- Comment by Victoria Gitman on Consistency strength of the existence of a transitive model of $\mathsf{ZFC}^-$ with a $\kappa$-complete ultrafilter
### Cantor’s Attic

- User:MkortinkUser account Mkortink was createdMkortink
- User:Julian Barathieu← Older revision Revision as of 11:23, 15 March 2018 Line 3: Line 3: For those interested, there is a discord server created by me and [[User:Zetapology|Zetapology]]: https://discord.gg/tbGrU39 For those interested, there is a discord server created by me and [[User:Zetapology|Zetapology]]: https://discord.gg/tbGrU39 −My "ideal background theory" (or just "prefered theory") is $\text{ZFC + GCH + […]Julian Barathieu

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# Category Archives: research

## The oddities of class forcing

I recently finished reading a series of excellent articles by Peter Holy, Regula Krapf, Philipp Lücke, Ana Njegomir and Philipp Schlicht investigating properties of class forcing over models of ${\rm GBC}$ (Gödel-Bernays set theory). So I would like to summarize … Continue reading

## Computable processes can produce arbitrary outputs in nonstandard models (continued)

In the previous post, I discussed a theorem of Woodin, extended recently by Blanck and Enayat, showing that for every computably enumerable theory $T$, there is in index $e$ such that if $M\models T$ satisfies that $W_e$ is contained in … Continue reading

## Set Theory Day to celebrate Joel David Hamkins’ 50th birthday

My former PhD supervisor, collaborator, and dear friend Joel David Hamkins is turning 50 this year. Evidently, Joel is much too young for a full-blown birthday conference, but some celebration must still be had. So to celebrate Joel’s birthday, we … Continue reading

## My first Black Board Day

Last Saturday, I participated in my first Black Board Day, an informal annual workshop dedicated to Gödel and organized for the past ten years by the neuroscientist Memming Park and his enthusiastic group of scientist friends.

## Separating the choice scheme from the parameter-free choice scheme in second-order arithmetic

This post is motivated by a really great paper of Wojciech Guzicki from the 1970s entitled “On weaker forms of choice in second-order arithmetic” [1] that I recently stumbled on while trying to trace the history of choice principles in … Continue reading

## Co-organizing the CUNY Set Theory Seminar

I was lucky enough to start graduate school and end up choosing the same adviser with two people who still remain my favorite colleagues and dearest friends. Coincidentally, the adviser also turned out to be quite good. The students were … Continue reading

Posted in personal, research
Tagged CUNY Set Theory Seminar, J. D. Hamkins, J. Reitz, T. Johnstone
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## An absoluteness lemma for countable embeddings

If $W$ is a transitive set or class and it thinks that there is an elementary embedding $j$ between some first-order structures $\mathcal M$ and $\mathcal N$, then $j$ is an actual elementary embedding and so the universe $V$ agrees … Continue reading

## Variants of Kelley-Morse set theory

Joel Hamkins recently wrote an excellent post on Kelley-Morse set theory (${\rm KM}$) right here on Boolesrings. I commented on the post about the variations one finds of what precisely is included in the ${\rm KM}$ axioms. I claimed that … Continue reading

## 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.