The next Set Theory in the UK workshop will take place online on Friday, 4 December 2020, from 9.30am-2pm.

Please see the meeting’s website http://www1.maths.leeds.ac.uk/~pmtadb/STUK6/STUK6.html for more information.

How to participate: Information how to obtain a login will be available on the conference website soon. Please find this information in advance, on the day before the meeting.

09.30-09.55 Yair Hayut: Generics via ultrapowers

10.00-10.50 Arno Pauly: Luzin’s (N) and randomness reflection

11.00-11.50 Peter Holy: Ramsey-like operators

lunch break

13.30-13.55 Jiachen Yuan: Indestructibility of supercompactness and large cardinals

Titles and abstracts:

Yair Hayut (Hebrew University of Jerusalem): Generics via ultrapowers

Bukovský and Dehornoy observed (independently) that there is a generic for the Prikry forcing over the iterated ultrapower by the measure. I will show how one can use this fact in order to derive (without referring to the forcing) many interesting properties of the generic extension.

Arno Pauly (Swansea University): Luzin’s (N) and randomness reflection

Peter Holy (University of Udine): Ramsey-like operators

Starting from measurability upwards, larger large cardinals are usually characterized by the existence of certain elementary embeddings of the universe, or dually, the existence of certain ultrafilters. However, below measurability, we have a somewhat similar picture when we consider certain embeddings with set-sized domain, or ultrafilters for small collections of sets. I will present some new results, and also review some older ones, showing that not only large cardinals below measurability, but also several related concepts can be characterized in such a way, and I will also provide a sample application of these characterizations.

Jiachen Yuan (University of East Anglia): Indestructibility of supercompactness and large cardinals

It is well known that “there is a supercompact cardinal which is immune to any $\kappa-$directed closed set forcing” is relatively consistent with “there is a supercompact cardinal”. We also know that there is no analogue of such a theorem to any large cardinal stronger than extendible. In fact, provably in $ZFC$ such large cardinal properties will be destroyed by any $\kappa-$directed closed set forcing. For larger cardinals, according to a theorem of Usuba, they can not survive in any set-forcing extension which is not equivalent to a small forcing. However, it was not known if it is possible to have such a large cardinal notion with its supercompactness indestructible. It turns out that this is true for a lot of large cardinals by forcing from a ground model with the same strength.

See you at the meeting!

Andrew Brooke-Taylor, Asaf Karagila and Philipp Schlicht