Projective determinacy for games of length $\omega^2$ and longer

Seminar
Speaker
Sandra Müller (KGRC)
Date
25/02/2019 - 15:00 - 13:00Add to Calendar 2019-02-25 13:00:00 2019-02-25 15:00:00 Projective determinacy for games of length $\omega^2$ and longer  We will study infinite two player games and the large  cardinal strength corresponding to their determinacy. For games of  length $\omega$ this is well understood and there is a tight  connection between the determinacy of projective games and the  existence of canonical inner models with Woodin cardinals. For games  of arbitrary countable length, Itay Neeman proved the determinacy of  analytic games of length $\omega \cdot \theta$ for countable $\theta  \> \omega$ from a sharp for $\theta$ Woodin cardinals. We aim for a converse at successor ordinals. In joint work with Juan  P. Aguilera we showed that determinacy of $\boldsymbol\Pi^1\_{n+1}$  games of length $\omega^2$ implies the existence of a premouse with  $\omega+n$ Woodin cardinals. This generalizes to a premouse with  $\omega+\omega$ Woodin cardinals from the determinacy of games of length  $\omega^2$ with $\Game^{\mathbb{R}}\boldsymbol\Pi^1\_1$ payoff. If time allows, we will also sketch how these methods can be adapted to, in combination with results of Nam Trang, obtain $\omega^\alpha+n$ Woodin cardinals for countable ordinals $\alpha$ and natural numbers $n$ from the determinacy of sufficiently long projective games. Building 105, Room 61 אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Building 105, Room 61
Abstract

 We will study infinite two player games and the large
 cardinal strength corresponding to their determinacy. For games of
 length $\omega$ this is well understood and there is a tight
 connection between the determinacy of projective games and the
 existence of canonical inner models with Woodin cardinals. For games
 of arbitrary countable length, Itay Neeman proved the determinacy of
 analytic games of length $\omega \cdot \theta$ for countable $\theta
 \> \omega$ from a sharp for $\theta$ Woodin cardinals.

We aim for a converse at successor ordinals. In joint work with Juan
 P. Aguilera we showed that determinacy of $\boldsymbol\Pi^1\_{n+1}$
 games of length $\omega^2$ implies the existence of a premouse with
 $\omega+n$ Woodin cardinals. This generalizes to a premouse with
 $\omega+\omega$ Woodin cardinals from the determinacy of games of length
 $\omega^2$ with $\Game^{\mathbb{R}}\boldsymbol\Pi^1\_1$ payoff.

If time allows, we will also sketch how these methods can be adapted
to, in combination with results of Nam Trang, obtain $\omega^\alpha+n$ Woodin
cardinals for countable ordinals $\alpha$ and natural numbers $n$ from
the determinacy of sufficiently long projective games.

תאריך עדכון אחרון : 19/02/2019