On December 1st 2016 I gave a talk in the KGRC Research Seminar.

**Abstract:** Let $x$ be a real of sufficiently high Turing degree, let $\kappa_x$ be the least inaccessible cardinal in $L[x]$ and let $G$ be $Col(\omega, {<}\kappa_x)$-generic over $L[x]$. Then Woodin has shown that $\operatorname{HOD}^{L[x,G]}$ is a core model, together with a fragment of its own iteration strategy.

Our plan is to extend this result to mice which have finitely many Woodin cardinals. We will introduce a direct limit system of mice due to Grigor Sargsyan and sketch a scenario to show the following result. Let $n \geq 1$ and let $x$ again be a real of sufficiently high Turing degree. Let $\kappa_x$ be the least inaccessible strong cutpoint cardinal of $M_n(x)$ such that $\kappa_x$ is a limit of strong cutpoint cardinals in $M_n(x)$ and let $g$ be $Col(\omega, {<}\kappa_x)$-generic over $M_n(x)$. Then $\operatorname{HOD}^{M_n(x,g)}$ is again a core model, together with a fragment of its own iteration strategy.

This is joint work in progress with Grigor Sargsyan.

Many thanks to Richard again for the great pictures!

*Related*

Maybe a stupid question, but what exactly is a core model in this context? Is it just a proper class extender model?

Hi Dan, I am happy that you found this announcement on my boolesrings page!

The title of my talk is inspired by John Steel’s and Hugh Woodin’s really nice paper “HOD as a core model”, published in the third cabal book. (Woodins result mentioned in the first paragraph of my abstract is published there.) What I mean here more precisely is, that this HOD is equal to the inner model $M_n(M_\infty, \Sigma)$, where $M_\infty$ is a direct limit model and $\Sigma$ is a fragment of its iteration strategy. Models like this are nowadays called “hod mice”. Btw, something like this is also on the conference T-shirt from Irvine (see here).

Ah I see, thanks! I do vaguely remember Steel defining the direct limit in his handbook article as well. I can’t wait to learn some more about this HOD business – thanks for the reference 🙂

Hey Sandra, can you give some details on the Turing degree that is needed?

Hey Stefan, to compute $\operatorname{HOD}^{M_n(x,g)}$ you want to consider reals $x$ which code $M_{n+1}|(\delta^{+\omega})^{M_{n+1}}$, where $\delta$ is the least Woodin cardinal in $M_{n+1}$. Then you can construct a direct limit system inside $M_n(x)$ as in Grigor Sargsyan’s paper “On the prewellorderings associated to the directed systems of mice”.

So there is no real mystery here, I was just trying to avoid some of the technical details in the abstract. 😉 But as its stated here you need to assume that $M_{n+1}$ exists as a tacit hypothesis.

Is the extender sequence allowed in the definition of HOD?

I just mean $\operatorname{HOD}$ in the universe of the model $M_n(x,g)$, as for example in the computation of $\operatorname{HOD}$ in models of determinacy.