Reflection principles and second-order choice principles with urelements

Yao, Bokai

doi:10.1016/j.apal.2021.103073

Cited by 3 publications

(3 citation statements)

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“…Let us begin with the observation that from supercompactness, we can produce models of urelement set theory with more than Ord many urelements, yet with second-order reflection. This observation is one of the core ideas and main results proved by the second author in [25,26], showing that if κ is κ + -supercompact, then there is a model of KMU with strictly more than Ord many urelements and the second-order reflection principle. Here, we generalize the observation to higher levels of supercompactness.…”

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

confidence: 69%

“…(Note that when w has size κ or larger, this is not the same as V κ (w), since w itself has rank 1 and would appear as a set, but in H κ (w) it would be a proper class.) The second author proved in [25] that KMU with at most Ord many urelements and second-order reflection is bi-interpretable with KM plus second-order reflection. The main contribution of this article, in contrast, will be to observe the dramatic increase in large-cardinal strength of second-order reflection when there are more than Ord many urelements.…”

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

confidence: 99%

“…Let us finally also address the central question left open in [25], concerning the strength of KMU with second-order reflection and "more than Ord many atoms." Yao had used a κ + -supercompact cardinal κ in order to produce a model of this theory, and the question is whether supercompactness is required, and specifically whether the large cardinal requirements will exceed the large cardinal notions consistent with V = L.…”

Section: Theorem 51 Every Model V ∈ V Of Gbc Is Bi-interpretable For ...mentioning

confidence: 99%

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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: 69%

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

confidence: 99%

Section: Theorem 51 Every Model V ∈ V Of Gbc Is Bi-interpretable For ...mentioning

confidence: 99%

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