Dear all, 

The next talks in the Bristol Logic and Set Theory Seminar will be given next Tuesday 23 May by Beatrice Pitton at 1.30-2.30pm and Bokai Yao at 3-4pm in room G2, Cotham House. 

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1.30pm

Speaker: Beatrice Pitton, University of Lausanne

Title: Definable subsets of the generalized Cantor and Baire spaces

Abstract: Generalized descriptive set theory (GDST) aims at developing a higher analogue of classical descriptive set theory in which ω is replaced with an uncountable cardinal κ in all definitions and relevant notions. In the literature on GDST it is often required that κ<κ = κ, a condition equivalent to κ regular and 2<κ = κ. In contrast, in this paper we use a more general approach and develop in a uniform way the basics of GDST for cardinals κ still satisfying 2<κ = κ but independently of whether they are regular or singular. This allows us to retrieve as a special case the known results for regular κ, but it also uncovers their analogues when κ is singular. We also discuss some new phenomena specifically arising in the singular context (such as the existence of two distinct yet related Borel hierarchies), and obtain some results which are new also in the setup of regular cardinals, such as the existence of unfair Borel∗ codes for all Borel∗ sets. This is joint work with Luca Motto Ros. 
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3pm 

Speaker: Bokai Yao, University of Notre Dame

Title: Forcing with Urelements

Abstract: I will begin by isolating a hierarchy of axioms based on ZFCU_R, which is ZFC set theory (with Replacement) modified to allow a class of urelements. For example, the Collection Principle is equivalent to the Reflection Principle over ZFCU_R, while it is folklore that neither of them is provable in ZFCU_R.
I then turn to forcing over countable transitive models of ZFU_R. A forcing relation is full just in case whenever a forcing condition p forces an existential statement, p also forces some instance of that statement. According to the existing approach, forcing relations are almost never full when there are urelements. I introduce a new forcing machinery to address this problem. I show that over ZFCU_R, the principle that every new forcing relation is full is equivalent to the Collection Principle. Furthermore, I show how forcing is able to preserve, destroy and resurrect the axioms in the hierarchy I introduced. In particular, the Reflection Principle is “necessarily forceble” in certain models of ZFCU_R. In the end, I will consider how the ground model definability can fail when the ground model contains a proper class of urelements.
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Upcoming Logic and Set Theory seminars will be announced at https://www.bristolmathsresearch.org/events/logic-and-set-theory. 

With best wishes, Philipp Schlicht, Kentaro Fujimoto and Philip Welch 

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