Saturday, May 10, 2014
Rowland Hall 306
Funded by NSF Grant DMS-1044150
2:00 – 3:00 Sherwood Hachtman (UCLA)
3:30 – 4:30 Bill Chen (UCLA)
5:00 – 6:00 Monroe Eskew (UCI)
It is a landmark result of Martin that Borel determinacy is a theorem of ZF. Interestingly, Martin’s inductive proof uses transfinitely many iterations of the Powerset axiom, and an analysis due to Friedman shows that these are necessary. In this talk, we will present a refinement of these results, giving level-by-level equiconsistencies between determinacy and a novel family of weak reflection principles. We will also discuss how these results adapt to the Borel hierarchy on coanalytic sets, where the inner model theory for measurable cardinals of high Mitchell order comes into play.
Mutual stationarity is a notion of stationarity for certain sequences of subsets of a singular cardinal $\lambda$ (possibly even of countable cofinality) which was defined by Foreman and Magidor. They isolated tight stationarity as a version of mutual stationarity that is easier to analyze. We use a pcf-theoretic scale to relate sequences of subsets of $\lambda$ to subsets of $\lambda^+$, translating from tightly stationary sequences to stationary subsets of $\lambda^+$. Then we will define careful sets, which are the subsets of $\lambda^+$ that are involved in this translation. The main result is the construction of a model where $\lambda$ is a strong limit and every subset of $\lambda^+$ is careful. This construction uses the combinatorics of tree-like scales and a diagonal supercompact Prikry forcing.
Abstract: We will discuss a “universal” forcing for collapsing a large cardinal to be the successor of smaller regular cardinal. It absorbs most of the effects of a wide class of “standard” collapsing posets, yet is incomparable with all of them, hence the name, anonymous collapse. Using this forcing, we show how to construct many different models with the same reals and same cardinals but very different cardinal characteristics of the continuum. We show how it can be used with almost-huge cardinals to achieve minimal solutions to Ulam’s measure problem, and to obtain successor cardinals which are “generically supercompact” in a strong sense. This will enable a solution to a question of Foreman related to an old conjecture in model theory about the cardinality of ultrapowers. In contrast to the traditional variety, we show that these generically supercompact cardinals are compatible with squares.