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

TL;DR

This paper shows that second-order reflection principles with abundant urelements in set theory are bi-interpretable with supercompact cardinals, establishing a deep connection between reflection and large cardinal axioms.

Contribution

It proves the bi-interpretability and equiconsistency of second-order reflection with abundant urelements and supercompact cardinals, using a novel reflection characterization of supercompactness.

Findings

Second-order reflection with abundant urelements is bi-interpretable with supercompact cardinals.

The proof uses a reflection characterization of supercompactness involving substructure truth transfer.

The results establish a precise equivalence between certain reflection principles and large cardinal assumptions.

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 $m\cap\kappa\in\kappa$.