14/Dec/2012: Trevor Wilson and Alex Rennet

14/December/2012, 11:o0–12:00
Fields institute,Room 230

Speaker: Trevor Wilson

Title: Well-behaved measures and weak covering for derived models

Abstract: For an inner model $M$ containing all the reals and
satisfying the Axiom of Determinacy, we show that countably complete
measures over $M$ on ordinals less than $\Theta^M$ are “well-behaved.”
In particular every such measure is ordinal-definable from $M$,
generalizing a theorem of Kunen that says “AD implies that every
measure on an ordinal less than Theta is ordinal-definable.” This
generalization is useful in constructing weak homogeneity systems
consisting of measures over $M$. As an application, we get a kind of
“weak covering” result that applies to weakly compact cardinal whose
successor is not computed correctly in HOD. Namely, if $\delta$ is a
weakly compact limit of Woodin cardinals, and $(\delta^+)^{\text{HOD}}
< \delta^+$, then the derived model at $\delta$ satisfies “every set
is Suslin.” The necessary facts about weak homogeneity systems,
Suslin sets, and derived models will all be covered in the talk.

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14/December/2012, 13:30–15:00
Fields institute,Room 230

Speaker: Alex Rennet

Title: Axiomatizability, Ultraproducts and O-Minimality
Abstract: Suppose a theory T is given as the set of sentences true in all structures in a fixed language which share some non-first order property P. For instance, if P is ‘stable’ or ‘o-minimal’ or ‘finite’, we get the L-theory of stability, o-minimality or finiteness respectively. A classic model-theoretic result describes the models of a theory constructed in this way as exactly those with the given property, or their ultraproducts (up to elementary equivalence).

The main result I’ll focus on in my talk is a general failure of recursive axiomatizability for certain theories of this kind. I’ll explain why the question of whether such a theory has a recursive axiomatization is natural, and give examples which do have such an axiomatization. In the case of o-minimality (a model-theoretic property related to non-standard analysis) this answers negatively a suggestion from the recent literature. I’ll go through the proof of this result, which pleasantly involves minimal model-theoretic detail.

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