Information: For the zoom access code, please contact Philip Welch in advance. Then, we characterise the recursive and admissible ordinals which are \Pi^1_2-soundness ordinals of some recursively enumerable extension of Arithmetical Comprehension. We go over the basic theory of |T| and use it to define the \Pi^1_2-soundness ordinal of T.
Time: Wednesday, 15 December, 16:30-18:00 UK time (17:30-19:00 CEST)Ībstract: We define the \Pi^1_2-norm |T| of a theory T, an analogue of the proof-theoretic ordinal of T for statements of complexity \Pi^1_2.
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Please contact Richard Springer for information how to participate. Information: This talk will be given via Zoom. We will discuss a recent generalization of their work to more general henselian valued fields, which is joint with Silvain Rideau-Kikuchi. In algebraically closed valued fields, the imaginaries were classified by Haskell-Hrushovski-Macpherson. In valued fields, the situation is more interesting, as there are definable quotients like the residue field and value group which may not be eliminated. In algebraically closed and in real closed fields, imaginaries may be eliminated. In the talk, we will discuss so-called imaginaries, i.e., definable quotients in first order logic. Title: Classification of definable quotientsĪbstract: In many areas of mathematics, quotient objects play an important role, and it is often useful to close a category under quotients. Speaker: Martin Hils, University of Münster Speaker: Todor Tsankov, Université Claude Bernard Lyon 1 Time: Thursday, 16 December, 01:00pm US central time (20:00 CET) Information: Join via the link on the seminar webpage
Finally, I will analyse and compare two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras, introducing new techniques to construct incomparable ultrafilters in this setting. Furthermore, I shall connect this discussion with a cardinal invariant of Boolean algebras, the ultrafilter number, and sketch consistency results (and open questions) concerning its possible values on Cohen and random algebras. First, I aim to introduce the framework of Tukey reducibility and discuss the existence of non-maximal ultrafilters. Title: Combinatorial properties of ultrafilters and their orderings on Boolean algebrasĪbstract: In this talk, I shall report on joint work with Jörg Brendle, focusing on the combinatorial properties of ultrafilters on Boolean algebras in relation to the Tukey and Rudin-Keisler orderings. Information: Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link. In this talk we will discuss iterating the C* construction, and show that under ZFC alone we can only reach finitely many steps, while a sequence of length $\omega$ is equiconsistent with an inner model with a measurable cardinal. This type of construction was considered with regards to HOD, where it was shown by McAaloon, Harrington, Jech and Zadrozny that “everything is possible” – on one hand there are models with iterated HODs of any ordinal length (and even of length Ord), and on the other hand it is possible that after \omega many stages the intersection either satisfies ZF without AC or even doesn’t satisfy ZF at all. that the result is no longer a model of ZFC). If this happens, one can iterate this construction, taking intersections at limit stages and ask at what stage it stabilizes or “breaks” (i.e. C* has some canonicity properties similar to L, but a notable difference is that C* does not necessarily satisfy the axiom V=C*, that is – constructing the C* of C* may result in a smaller inner model.
Title: Iterating the cofinality-\omega constructible modelĪbstract: Consider C* – the inner model constructed in an L-like fashion, but using first order logic augmented with the “cofinality \omega” quantifier. Hebrew University-Bar Ilan University Set Theory seminar The talks are listed in reverse chronological order, so please scroll down to find today’s talks!
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