Dissertation. PDF

- S. Uhlenbrock.
*Pure and Hybrid Mice with Finitely Many Woodin Cardinals from Levels of Determinacy*. PhD thesis, WWU Münster, 2016.`@phdthesis{Uh16, author = {S. Uhlenbrock}, title = {{Pure and Hybrid Mice with Finitely Many Woodin Cardinals from Levels of Determinacy}}, school = {WWU Münster}, year = 2016 }`

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 $M_n^\#(x)$ 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 $M_n^\#(x)$ 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so far unpublished result by W. Hugh Woodin.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 $M_k^{\Sigma, \#}$.

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.