- 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}, }`

Download pdf at arXiv:2204.09766

**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

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