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REFLECTION IN SECOND-ORDER SET THEORY WITH ABUNDANT URELEMENTS BI-INTERPRETS A SUPERCOMPACT CARDINAL
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 23 December 2022, pp. 1007-1043
- Print publication:
- September 2024
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- Article
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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 that 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 $.
