Fri Jan 22, 2016
60 Evans Hall, 4–6 PM
John Steel (UC Berkeley)
Ordinal Definability in Models of the Axiom of Determinacy
Let HOD be the class of all hereditarily ordinal definable sets, and let M be a model of the Axiom of Determinacy. The inner model HOD^M consisting of all sets that are hereditarily ordinal definable in the sense of M is of great interest, for a variety of reasons. In the first part of the talk, we shall give a general introduction that explains some of these reasons.
We shall then describe a general Comparison Lemma for “hod mice” (structures that approximate HOD^M). Modulo still-open conjectures on the existence of iteration strategies, this lemma yields models M of the Axiom of Determinacy such that HOD^M can be analyzed fine structurally (for example, satisfies the GCH), and yet satisfies very strong large cardinal hypotheses (for example, that there are superstrong cardinals).