## Special Colloquium

Dima Sinapova

University of California at Irvine

The singular cardinal problem

**Abstract:**Cardinal arithmetic has a long history, dating back to Cantor. After the invention of forcing, any reasonable behavior of regular cardinals was shown to be consistent. In contrast, results about singular cardinals are much more intricate and require more than the standard axioms of set theory. I will discuss relative consistency results about singular cardinal arithmetic in the context of forcing and large cardinals. I will focus on the relationship between the Singular Cardinal Hypothesis, which is a parallel of the Continuum Hypothesis for singular cardinals, and combinatorial principles such as Jensen's square principle, scales, and the tree property. I will describe some recent consistency results about singular combinatorics and new constructions involving Prikry type forcings.

Monday January 30, 2012 at 3:00 PM in SEO 636