Generalized Baire spaces and closed Maximality Principles

Set theory seminarFriday, March 20, 201510:00 amGC 6417

Philipp Lücke

Generalized Baire spaces and closed Maximality Principles

Rheinische Friedrich-Wilhelms-Universität Bonn

Given an uncountable regular cardinal $\kappa$, the generalized Baire space of $\kappa$ is set ${}^\kappa\kappa$ of all functions from $\kappa$ to $\kappa$ equipped with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $\kappa$.
A subset of this space is $\mathbf{\Sigma}^1_1$ (i.e. a projection of a closed subset of ${}^\kappa\kappa\times{}^\kappa\kappa$) if and only it is definable over $\mathrm{H}(\kappa^+)$ by a $\Sigma_1$-formula with parameters. This shows that the class of $\mathbf{\Sigma}^1_1$-subsets contains a great variety of set-theoretically interesting objects. Moreover, it is known that many basic and interesting questions about sets in this class are not decided by the axioms of $\mathrm{ZFC}$ plus large cardinal axioms.

In my talk, I want to present examples of extensions of $\mathrm{ZFC}$ that settle many of these questions by providing a nice structure theory for the class of $\mathbf{\Sigma}^1_1$-subsets of ${}^\kappa\kappa$. These forcing axioms appear in the work of Fuchs, Leibman, Stavi and Väänänen. They are variations of the maximality principle introduced by Stavi and Väänänen and later rediscovered by Hamkins.

Dr. Philipp Lücke is currently a post-doctoral fellow at the University of Bonn. He holds a PhD from the University of Münster, completed under the supervision of Ralf Schindler in 2012. His research interests include set theory and applications of set theory in algebra.

Posted by on February 28th, 2015