## Downward transference of mice and universality of local core models

April 4, 2012

Martin Zeman and I have just submitted our paper Downward transference of mice and universality of local core models to the Journal of Symbolic Logic, downloadable from my papers page. We have also uploaded it on the ArXiv. (I should have been doing this for years; this is the first time I post there.)

It is a nice observation that goes back to Friedman that if $0^\sharp$ exists and ${\mathbf M}$ is an inner model that correctly computes $\omega_2$, then $0^\sharp\in {\mathbf M}$. Looking at a completely different problem, from the theory of forcing axioms, we were led to the question of how much this result can be generalized.

Our main result is that there is a significant transfer of structure going on, simply due to the agreement of cardinals. (The statement of the result and, of course, the argument, require familiarity with fine structure theory, as developed in Steel’s or Martin’s books.)

Theorem. Assume that ${\mathbf M}$ is a proper class inner model, and that $\delta$ is regular in ${\mathbf V}$.

1. If there are no inner models of ${\mathbf V}$ with Woodin cardinals, $\delta>\omega_1$, and

$\{x\in{\mathcal P}_\delta(\delta^+)\cap{\mathbf M}\mid{\rm cf}^{\mathbf M}(x\cap\delta)>\omega\}$

is stationary, then ${\mathbf K}^{\mathbf M}\|\delta$ is universal for all iterable 1-small premice in ${\mathbf V}$ of cardinality less than $\delta$.

2. If, in ${\mathbf M}$, $0^\P$ does not exist, and ${\mathcal P}_\delta(\delta^+)\cap{\mathbf M}$ is stationary, and $\delta>\omega_1$, then the same conclusion holds. If $\delta=\omega_1$, then ${\mathbf K}^{\mathbf M}\|\omega_2$ is universal for all countable iterable premice in ${\mathbf V}$.

Here, as usual, ${\mathcal P}_\kappa(\lambda)$ denotes the collection of subsets of $\lambda$ of size less than $\kappa$. It is easy to check that ${\mathcal P}_{\omega_1} (\omega_2)\cap{\mathbf M}$ is stationary if ${\mathbf M}$ computes $\omega_2$ correctly, so this result generalizes the statement about $0^\sharp$ mentioned above.

In fact, it follows that, if $\omega_2$ is computed correctly in ${\mathbf M}$, then any sound mouse in ${\mathbf V}$ projecting to $\omega$ and below $0^\P$, is in ${\mathbf M}$. Beyond $0^\P$, the argument becomes more complicated, and we need to assume a global anti-large cardinal assumption, namely, that there are no inner models in ${\mathbf V}$ with Woodin cardinals.

We expect that this restriction can be weakened, perhaps even dispensed with.

(This paper is the second in a series, aiming to explore the structure of inner models for which some agreement of cardinals holds. I briefly mentioned the first paper here.)