The surprising strength of second-order reflection in urelement set theory, Luminy, October 2023

This will be a talk at the XVII International Luminy Workshop in Set Theory at the Centre International de Rencontres Mathématiques (CIRM) near Marseille, France, held 9-13 October 2023.

Abstract. I shall give a general introduction to urelement set theory and the role of the second-order reflection principle in second-order urelement set theory GBCU and KMU. With the abundant atom axiom, asserting that the class of urelements greatly exceeds the class of pure sets, the second-order reflection principle implies the existence of a supercompact cardinal in an interpreted model of ZFC. The proof uses a reflection characterization of supercompactness: a cardinal is supercompact if and only if for every second-order sentence $\psi$ true in some structure $\langle M,\ldots\rangle$ (of any size) in a language of size less than $\kappa$ is also true in a first-order elementary substructure $m\prec M$ of size less than $\kappa$ with $m\cap\kappa\in\kappa$. This is joint work with Bokai Yao.

Set theory with abundant urelements, STUK 10, Oxford, June 2023

This will be a talk for the Set Theory in the UK, STUK 10, held in Oxford 14 June 2023, organized by my students Clara List, Emma Palmer, and Wojciech Wołoszyn.

Abstract. I shall speak on the surprising strength of the second-order reflection principle in the context of set theory with abundant urelements. The theory GBcU with the abundant urelement axiom and second-order reflection is bi-interpretable with a strengthening of KM with a supercompact cardinal. This is joint work with Bokai Yao.