STUK 6:

The sixth meeting of the LMS Set Theory in the UK network will be held online on Friday the 4th of December, 2020. Talks will be given by Arno Pauly, Peter Holy, Yair Hayut and Jiachen Yuan, to which all are welcome.

We will also end the day with a series of informal minitalks, particularly encouraging local graduate students to introduce themselves to the STUK community if they are new, and to talk about what they are working on.

The organisers would like to apologise for the fact that none of the invited speakers are women. In the scramble to arrange this meeting at the last minute, insufficient consideration was given to the diversity of the speakers.

Coordinates and Contact

The meeting will happen via Zoom, at this address:
https://us02web.zoom.us/j/83344946660

Meeting ID: 833 4494 6660
The password is a ten-letter word starting with "f", all lower case, and is the name of the ZFC axiom which guarantees, in particular, that no set can be an element of itself.

If you would like to volunteer a mini-talk, or need to contact the organisers for any other reason, contact Andrew Brooke-Taylor (a.d.brooke-taylor@bla.comleeds.ac.uk), Asaf Karagila (karagila@bla.commath.huji.ac.il), and/or Philipp Schlicht (philipp.schlicht@bla.combristol.ac.uk).

Schedule

All times are UK (GMT).

Abstracts

Generics via ultrapowers

Bukovsky 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.

Luzin's (N) and randomness reflection

Arno spoke about material from his new paper with Westrick and Yu, arXiv:2006.07517.

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.

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.

Previous meetings

• STUK 0 (preliminary meeting): The Royal Society, London, 7 November 2018
• STUK 1: University of Cambridge, 16 February 2019
• STUK 2: University of Bristol, 8 May 2019
• STUK 3: University of Leeds, 3 October 2019
• STUK 4: University of Oxford, 14 December 2019
• STUK 5: The Royal Society, London, 11 February 2020