Subcompact Cardinals, Type Omission, and Ladder Systems

Hayut, Yair; Magidor, Menachem

doi:10.1017/jsl.2022.11

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In the Interpreted Model V [[ ]] Of Zfcu With Many Urelement...1

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2022

2024

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“…Theorem 8.4 thus provides a more refined analysis than Theorem 8.2 in the characterization of the cardinals that are -reflective for Π 1 1 assertions, which are exactly the nearly -supercompact cardinals. This theorem is part of a constellation of closely related, similar results in the research literature, including [2, Theorems 3.5 and 4.7], [3,Theorem 1.4], and [12,Lemma 2.8].…”

Section: In the Interpreted Model V [[ ]] Of Zfcu With Many Urelement...mentioning

confidence: 55%

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

Hamkins¹,

Yao²

2022

J. symb. log.

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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 $ .

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Section: In the Interpreted Model V [[ ]] Of Zfcu With Many Urelement...mentioning

confidence: 55%