## Luminy – Coda

October 27, 2010

While at Luminy, David Asperó showed me a quick proof of a nice result on Reinhardt cardinals in ${\sf ZF}$. It complements Grigor Sargsyan’s result discussed here.

Theorem (Asperó). Work in ${\sf ZF}$. Suppose $j:V\to V$ is a nontrivial elementary embedding. Then there are a $\bar\kappa<{\rm cp}(j)$ and an ordinal $\alpha$ such that for all $\beta$ there is a $\mu$ and an elementary $\pi:V_\mu\to V_\mu$

such that ${\rm cp}(\pi)=\bar\kappa$ and $\pi(\alpha)>\beta$.

Proof. For $\alpha$ an ordinal, set $\kappa^\alpha=\min\{\kappa\mid\exists\mu\exists i:V_\mu\to V_\mu$ such that ${\rm cp}(i)=\kappa$ and ${\rm ot}(\{\beta<\mu\mid i(\beta)=\beta\})\ge\alpha\}$.

Note that suitable fragments of $j$ witness that $\kappa^\alpha$ is defined for all $\alpha$. Moreover, $\alpha<\beta$ implies that $\kappa^\alpha\le\kappa^\beta\le{\rm cp}(j)$, and therefore there is a $\bar\kappa\le{\rm cp}(j)$ such that $\kappa^\beta=\bar\kappa$ for all $\beta$ sufficiently large. Moreover, since it is definable, we actually have $\bar\kappa<{\rm cp}(j)$.

Let $\alpha$ be least with $\kappa^\beta=\bar\kappa$ for $\beta\ge\alpha$. We claim that $\bar\kappa$ and $\alpha$ are as wanted. For this, consider some $\beta>\alpha$, and pick $i:V_\mu\to V_\mu$ witnessing that $\bar\kappa=\kappa^\beta$. All we need to do is to check that $i(\alpha)\ge\beta$.

But note that if $\gamma\in[\alpha,\beta)$, then $V_\mu\models\kappa^\gamma=\bar\kappa$ Hence, if $i(\alpha)<\beta$, we have $V_\mu\models \kappa^{i(\alpha)}=\bar\kappa$.

But $\kappa^{i(\alpha)}=i(\kappa^\alpha)=i(\bar\kappa)>\bar\kappa$. Contradiction. $\Box$

## Luminy – Hugh Woodin: Ultimate L (III)

October 27, 2010

For the first lecture, see here.

For the second lecture, see here.

## Luminy – Hugh Woodin: Ultimate L (II)

October 21, 2010

For the first lecture, see here.

## Luminy – Hugh Woodin: Ultimate L (I)

October 19, 2010

The XI International Workshop on Set Theory took place October 4-8, 2010. It was hosted by the CIRM, in Luminy, France. I am very glad I was invited, since it was a great experience: The Workshop has a tradition of excellence, and this time was no exception, with several very nice talks. I had the chance to give a talk (available here) and to interact with the other participants. There were two mini-courses, one by Ben Miller and one by Hugh Woodin. Ben has made the slides of his series available at his website.

What follows are my notes on Hugh’s talks. Needless to say, any mistakes are mine. Hugh’s talks took place on October 6, 7, and 8. Though the title of his mini-course was “Long extenders, iteration hypotheses, and ultimate L”, I think that “Ultimate L” reflects most closely the content. The talks were based on a tiny portion of a manuscript Hugh has been writing during the last few years, originally titled “Suitable extender sequences” and more recently, “Suitable extender models” which, unfortunately, is not currently publicly available.

The general theme is that appropriate extender models for supercompactness should provably be an ultimate version of the constructible universe $L$. The results discussed during the talks aim at supporting this idea.