# Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal

• J. D. Hamkins and B. Yao, “Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal,” Mathematics arXiv, 2022.
[Bibtex]
@ARTICLE{HamkinsYao:Reflection-in-second-order-set-theory-with-abundant-urelements,
author={Joel David Hamkins and Bokai Yao},
year={2022},
eprint={2204.09766},
archivePrefix={arXiv},
primaryClass={math.LO},
title = {Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal},
journal = {Mathematics arXiv},
volume = {},
number = {},
pages = {},
month = {},
note = {manuscript under review},
abstract = {},
keywords = {},
source = {},
doi = {},
url = {http://jdh.hamkins.org/second-order-reflection-with-abundant-urelements},
}

Abstract. After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $\kappa$ is supercompact if and only if every $\Pi^1_1$ sentence true in a structure $M$ (of any size) containing $\kappa$ in a language of size less than $\kappa$ is also true in a substructure $m\prec M$ of size less than $\kappa$ with $m\cap\kappa\in\kappa$.

See also my talk at the CUNY Set Theory Seminar: The surprising strength of reflection in second-order set theory with abundant urelements

# The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022

This was an online talk 15 April 12:15 for the CUNY Set Theory Seminar. Held on Zoom at 876 9680 2366.

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 $\kappa$ is supercompact if and only if for every second-order sentence $\psi$ true in some structure $M$ (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 $\kappa$. This is joint work with Bokai Yao.

# The surprising strength of reflection in second-order set theory with abundant urelements, Konstanz, December 2021

This will be talk for the workshop Philosophy of Set Theory held at the University of Konstanz, 3 – 4 December 2021 — in person!

Update: Unfortunately, the workshop has been cancelled (perhaps postponed to next year) in light of the Covid resurgence.

Abstract. I shall analyze the roles and interaction of reflection and urelements in second-order set theory. Second-order reflection already exhibits large cardinal strength even without urelements, but recent work shows that in the presence of abundant urelements, second-order reflection is considerably stronger than might have been expected—at the level of supercompact cardinals. This is joint work with Bokai Yao (Notre Dame).