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

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so far unpublished result by W. Hugh Woodin.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.