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Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it is possible to characterize them using inner model theory. For instance the $HOD^{L(\mathbb{R})}$ is a certain mouse, $\mathcal M_{\omega}$, which has information about its iteration strategy, namely it is the direct limit of a directed system containing all mice which are properly small, full and are $\Game^{\mathbb{R}} \Pi^1_1$-iterable (this last condition turns out to be the same as $\Sigma_1^{L(\mathbb{R})}$ and these statements are absolute between $V$ and $L(\mathbb{R})$ by Martin and Steel, see Cabal books).

Can we similarly characterize all of $L(\mathbb{R})$ as a certain type of mouse? More generally can one characterize models of $AD$ by just saying that they are the mice satisfying some basic properties? I am reading the CMIP of Steel (i.e I'm still stuck in the 90's) so I don't know how far Core Models have reached today but can one construct a Core Model $K$ for $\omega$ many Woodin Cardinals and a measurable above them and then somehow derive another core model from it which would satisfy determinacy? Thx.

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up vote 4 down vote accepted

Certainly, $L(\mathbb R)$ is not a mouse (rather, a weasel) over a countable set, and the only way I see of thinking of it as a mouse and still capturing all the reals is making it a mouse over $\mathbb R$, in which case the answer is trivial. There is interesting work on $\mathbb R$-premice, but I do not think this is what you had in mind here.

More relevantly, if determinacy holds, then $L(\mathbb R)$ is a derived model, which I think is how you should think of it. There is a key relationship between models of determinacy and models with limit many Woodin cardinals, given by the derived model theorem. This is significant result. It says, on the one hand, that a symmetric extension of any model of choice with limit many Woodin cardinals results in a model of $\mathsf{AD}^+$. On the other hand, by an appropriate use of Prikry forcing, one can weave together $\mathsf{HOD}$-like models inside models of $\mathsf{AD}^+$, and recover a model of choice with limit many Woodin cardinals. This is how several of the key equiconsistencies between versions of determinacy and appropriate instances of large cardinals have been proved.

Derived models of (fine-structural) mice are special, of course, and there is a significant body of work on their properties. (And I think we can still say much more.)

A good reference for both topics ($\mathbb R$-premice and derived models of mice) is the paper

John R. Steel. Derived models associated to mice. In Computational prospects of infinity. Part I. Tutorials, pp. 105–193, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 14, World Sci. Publ., Hackensack, NJ, 2008. MR2449479 (2009m:03082).

(A preprint is available at John's page, where you will also find preprints of papers presenting the derived model construction in detail.)

Work on $\mathbb R$-premice leads to the notion of $K(\mathbb R)$, which indeed gives us a nontrivial way of thinking of some models of determinacy as mice, as we can think of "nice" models of $\mathsf{AD}^+$ as initial segments of $K(\mathbb R)$. (The precise meaning of niceness here is technical, and probably not too illuminating at this point. I mean that appropriate versions of the mouse set conjectures hold. John's paper covers this carefully.)

Work on the core model induction gives us that not just from large cardinals, but also from many strong combinatorial statements, we get many mice (over countable sets) inside $L(\mathbb R)$ (or inside general models of determinacy), and the fact that determinacy holds in $L(\mathbb R)$ can be thought of as a consequence of the presence of these mice. The key notion is the concept of a mouse operator, and a good introduction to this topic is the beginning of the monograph in preparation by John and Ralf:

Ralf Schindler and John R. Steel. The core model induction.

For the gap between CMIP and these papers, you may want to look at John's paper in the Handbook, a preprint of which can also be found at his page:

John R. Steel. An outline of inner model theory. In Handbook of set theory. Vols. 1, 2, 3, pp. 1595–1684, Springer, Dordrecht, 2010. MR2768698.

You are probably interested in this general area, so you may enjoy knowing that we posted notes of many of the talks and background material at the page for the first conference on the core model induction and hod mice, Münster, July 19-August 06, 2010.

For the mouse set conjectures in particular, you want to read Grigor's thesis (posted at the conference site), or the preprint he wrote based on it, and available at his webpage:

Grigor Sargsyan. A tale of hybrid mice.

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Thank you Andres. That's a really neat answer as usual. Indeed, the first part of my question was trivial, I didn't realize this, $L(\mathbb{R})$ is just a mouse over $\mathbb{R}$. I have all these references next to me. After having read the FSIT, Steve's and John's articles in the handbook, half of the CMIP, and some Cabal stuff, I didn't know exactly where to go here. In any case, Descriptive set theory and inner model theory, when put together, become a unified fascinating subject. I guess I am going to look at the derived models associated to mice now. – Carlo Von Schnitzel Mar 14 '13 at 22:07

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