## Set theory in the UK

*Set Theory in the UK*is a joint research group in set theory with members at the Universities of Bristol, Cambridge, East Anglia, Leeds, Oxford, Warwick and University College London. A preliminary meeting of the joint research group was held at the Royal Society in London on 7 November 2018. The first meeting took place Cambridge on 16 February 2019.

The second meeting took place on Wednesday, 8 May 2019, 11.00-18.00, in the School of Mathematics, University of Bristol, Howard House, Queen's Avenue, Bristol BS8 1SD. It was organized by Dan Nielsen, Philipp Schlicht and Philip Welch. The meeting was partially funded by a London Mathematical Society Scheme 3 grant and Marie Curie Individual Fellowship 794020. It was open to everyone.

### Slides and open research questions

Slides by Victoria Gitman.List of open research problems presented in the talks, discussions and open problems sessions.

### Program

11.00-12.00**Set theory in second-order**(Victoria Gitman, CUNY)

12.00-13.00 Lunch at the math department

13.00-14.00

**Set theory and category theory**(Andrew Brooke-Taylor, University of Leeds)

14.00-15.00 Presentations of open problems: Asaf Karagila, Dan Nielsen, Philip Welch

15.00-16.00 Discussions

16.00-16.30 Coffee and cake

16.30-18.00 Discussions

The talks were introductory and lead to discussions of open questions in the afternoon. There was ample time for collaboration and discussions.

### Abstracts

Andrew Brooke-Taylor: Set theory and category theoryLarge cardinal axioms have been used in category theory for decades, particularly in the study of

*accessible categories*, which are a framework for a kind of category-theoretic model theory. I will give an introduction to this area, from basic definitions up to recent results.

Victoria Gitman: Set theory in second-order

Classes, from class forcing notions to elementary embeddings of the universe to inner models, play a fundamental role in modern set theory. But within first-order set theory we are limited to studying only definable classes and we cannot even express properties that necessitate quantifying over classes. Second-order set theory is a formal framework in which a model consists both of a collection of sets and a collection of classes (which are themselves collections of sets). In second-order set theory, we can study classes such as truth predicates, which can never be definable over a model of ZFC, and properties that, for instance, quantify over all inner models. With this formal background we can develop a theory of class forcing that explains why and when class forcing behaves differently from set forcing. In this talk, I will discuss a hierarchy of second-order set theories, starting from the weak Gödel-Bernays set theory GBC and going beyond the relatively strong Kelley-Morse theory KM. I will give an overview of a number of interesting second-order set theoretic principles that arose out of recent work in this area, such as, class choice principles, transfinite recursion with classes, determinacy of games on the ordinals, and the class Fodor Principle. The study of where these principles fit in the hierarchy of second-order set theories should serve as the beginning of a reverse mathematics program that I hope this talk will encourage set theorists to take part in.

### Participants

Bea Adam-Day, LeedsJoe Allen, Bristol

Hazel Brickhill, Bristol

Andrew Brooke-Taylor, Leeds

Catrin Campbell-Moore, Bristol

Adam Epstein, Warwick

Kentaro Fujimoto, Bristol

Victoria Gitman, New York

John Howe, Leeds

Asaf Karagila, Norwich

Chris Le Sueur, Norwich

Richard Matthews, Leeds

Dan Nielsen, Bristol

Simon Peacock, Bristol

Jeremy Rickard, Bristol

Philipp Schlicht, Bristol

Johannes Stern, Bristol

Christopher Turner, Bristol

Philip Welch, Bristol