### Recent Writing

- Virtual large cardinal principles
- Filter games and Ramsey-like cardinals
- The exact strength of the class forcing theorem
- A model of the generic Vopěnka principle in which the ordinals are not $\Delta_2$-Mahlo
- A model of second-order arithmetic with the choice scheme in which $\Pi^1_2$-dependent choice fails

### Mathoverflow Activity

- Comment by Victoria Gitman on Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?@Stefan For some reason, I cannot click on the link.Victoria Gitman
- Comment by Victoria Gitman on Cofinality of $j(\kappa)$ for a measurability embedding $j:V\to M$ with critical point $\kappa$@AsafKaragila I edited the question to make it clearer. Maybe I will ask your version as a follow-up if you don't ask it first :).Victoria Gitman
- Comment by Victoria Gitman on Cofinality of $j(\kappa)$ for a measurability embedding $j:V\to M$ with critical point $\kappa$@AsafKaragila I should have done a better job stating my question clearly. I wanted to know precisely what Yair answered: whether there are models where $\text{cf}(j(\kappa))$ is smaller than the size of $j(\kappa)$. Your interpretation of the question is very interesting as well, but I think much harder to answer.Victoria Gitman

- Comment by Victoria Gitman on Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?
### Cantor’s Attic

- Constructible universeImplications, equivalences, and consequences of $0^\#$'s existence ← Older revision Revision as of 11:26, 14 December 2017 Line 63: Line 63: If $0^\#$ exists then: If $0^\#$ exists then: −* $\aleph_\omega$ is [[stable]] in $L$ and so $0^\#$ also corresponds to the set of the Gödel numberings of first-order formulas $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$+* $L_{\aleph_\omega}\prec […]Julian Barathieu
- Axiom of determinacy← Older revision Revision as of 11:13, 14 December 2017 Line 100: Line 100: * Every uncountable cardinal $Julian Barathieu
- Upper attic← Older revision Revision as of 11:07, 14 December 2017 Line 30: Line 30: * [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal * [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal * '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal * '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal −* [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, '''[[Ramsey]]''' cardinal, [[strongly Ramsey]] […]Julian Barathieu
- Strongly compactDiverse characterizations ← Older revision Revision as of 08:57, 14 December 2017 Line 18: Line 18: A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an [[elementary embedding]] $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some […]Julian Barathieu
- Supercompact← Older revision Revision as of 21:12, 11 December 2017 Line 10: Line 10: One can see the equivalence of the two formulations by first considering the ultrafilter $U$ arising from the [[seed]] $j''\theta$, so that $X\in U\iff j''\theta\in j(X)$. It is easy to check that $U$ is a normal fine measure on $\mathcal{P}_\kappa(\theta)$. Conversely, […]Julian Barathieu

- Constructible universe

# 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
4 Comments

## 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.