Dissertation. PDF

Uhlenbrock, S. (2016). *Pure and Hybrid Mice with Finitely Many Woodin Cardinals from Levels of Determinacy* (PhD thesis). WWU Münster.

Mice are sufficiently iterable canonical models of set theory. Martin and Steel showed in the 1980s that for every natural number $n$ the existence of $n$ Woodin cardinals with a measurable cardinal above them all implies that boldface $\boldsymbol\Pi^1_{n+1}$ determinacy holds, where $\boldsymbol\Pi^1_{n+1}$ is a pointclass in the projective hierarchy. Neeman and Woodin later proved an exact correspondence between mice and projective determinacy. They showed that boldface $\boldsymbol\Pi^1_{n+1}$ determinacy is equivalent to the fact that the mouse $M_n^\sharp(x)$ exists and is $\omega_1$-iterable for all reals x.

We prove one implication of this result, that is boldface $\boldsymbol\Pi^1_{n+1}$ determinacy implies that $M_n^\sharp(x)$ exists and is $\omega_1$-iterable for all reals $x$, which is an old, so far unpublished result by W. Hugh Woodin. As a consequence, we can obtain the determinacy transfer theorem for all levels $n$.

Following this, we will consider pointclasses in the $L(\mathbb{R})$-hierarchy and show that determinacy for them implies the existence and $\omega_1$-iterability of certain hybrid mice with finitely many Woodin cardinals, which we call $M_k^{\Sigma, \sharp}$. These hybrid mice are like ordinary mice, but equipped with an iteration strategy for a mouse they are containing, and they naturally appear in the core model induction technique.