Ordinal definability in extender models

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Abstract

Gödel’s universe L of constructible sets admits a detailed analysis, and ZFC decides much of its first-order theory. This makes the study of L tractable and interesting. However, many natural, desirable set theoretic principles – in particular, moderate strength large cardinal and determinacy principles – must fail in L. Extender models L[E] are generalizations of L, which still admit a detailed analysis, but can also satisfy large cardinal principles. The predicate E is a sequence of extenders, which witness large cardinals. The more canonical L[E] are called iterable; iterability is witnessed in V by an iteration strategy. It is known that if L[E] has Woodin cardinals then it cannot know too much of this iteration strategy. One can ask whether E can be defined over L[E], possibly from some parameter. Related to this, one can ask about the structure of HODL[E] (that is, the universe HOD of hereditarily ordinal definable sets, as computed in L[E]). I will survey what is known to the author regarding these questions. In simple cases, L[E] satisfies “V=HOD”. This holds in L, and, for example, in the minimal proper class L[E] with a measurable cardinal. But in the presence of Woodin cardinals, the question becomes harder to understand. I will cover the following recent results. Let L[E] be iterable and satisfy ZFC. Then (i)  E is definable from the parameter E↾ω_1^L[E] in L[E]; and (ii) in many circumstances, L[E] is a small forcing extension of HOD^L[E] and HOD^L[E] admits a detailed analysis above ω_2^L[E]. However, the full structure of HOD^L[E] is an open question, even in rather basic cases.

Speakers from the University of Münster

Schlutzenberg, Farmer

Junior professorship for mathematical logic (Prof. Schlutzenberg)