Talk in the KGRC Research Seminar

On June 9th 2016 I gave a talk in the KGRC Research Seminar in Vienna.

Hybrid Mice and Determinacy in the $L(\mathbb{R})$-hierarchy.

Abstract: This talk will be an introduction to inner model theory the at the
level of the $L(\mathbb{R})$-hierarchy. It will
focus on results connecting inner model theory to the determinacy of
certain games.

Mice are sufficiently iterable models of set theory. Martin and Steel
showed in 1989 that the existence of finitely many Woodin cardinals
with a measurable cardinal above them implies that projective
determinacy holds. Neeman and Woodin proved a level-by-level
connection 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 $x$.

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, which enables them to capture certain sets
of reals. We will discuss what it means for a mouse to capture a set
of reals and outline why hybrid mice fulfill this task. If time allows we will sketch a proof that determinacy for sets of reals in the $L(\mathbb{R})$-hierarchy implies the existence of hybrid mice.

Many thanks to Richard for the pictures!

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