Event Detail
Fri Jan 22, 2016 60 Evans Hall, 4–6 PM |
Logic Colloquium 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).