Workshop on Set Theory
(18 - 22 Jul 2011)

Jointly funded by the John Templeton Foundation

~ Abstracts ~

Dichotomies for analytic k-gaps
Stevo Todorcevic, University of Toronto, Canada


This is a joint work with Antonio Aviles. We give higher-dimensional versions of the analytic gap dichotomies introduced by the author some fifteen years ago. This leads us to a new classification theory for analytic k-gaps.

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Harmonic analysis, overspill, forcing
Jindrich Zapletal, University of Florida, USA


A sigma-ideal I on a compact metric space is said to have an overspill property if there is no analytic set of compact sets in I that contains all countable compact sets. This classical descriptive set theoretic notion in fact corresponds to a certain forcing preservation property of the quotient poset of I-positive Borel sets. As an example, I will discuss the sigma-ideal generated by H-sets from harmonic analysis. The quotient forcing is bounding and in fact close to a forcing introduced by Shelah twenty years ago. The overspill property holds.

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Generalizations of the Kunen inconsistency
Joel Hamkins, The City University of New York, USA


I shall present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself, including generalizations-of-generalizations previously established by Woodin and others. For example, there is no nontrivial elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or from V to the gHOD, or conversely from gHOD to V; indeed, there can be no nontrivial elementary embedding from any definable class to V. Other results concern generic embeddings, definable embeddings and results not requiring the axiom of choice. I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions. This is joint work with Greg Kirmayer and Norman Perlmutter.

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