Reflection in Second-Order Set Theory with Abundant Urelements Bi-Interprets a Supercompact Cardinal

Journal of Symbolic Logic:1-36 (forthcoming)
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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, 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 κ is supercompact if and only if every Π11 sentence true in a structure M (of any size) containing κ in a language of size less than κ is also true in a substructure m≺M of size less than κ with m∩κ∈κ.



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Author Profiles

Bokai Yao
University of Notre Dame
Joel David Hamkins
Oxford University

Citations of this work

Reflective Mereology.Bokai Yao - forthcoming - Journal of Philosophical Logic:1-26.

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