Pure and Hybrid Mice with Finitely Many Woodin Cardinals from Levels of Determinacy

Dissertation. PDF. Bibtex.

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determinacy implies that $M_n^\sharp(x)$ exists and is $\omega_1$-iterable for all reals $x$, which is an old, so far unpublished result by W. Hugh Woodin.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.