Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal
| Autor: | Hamkins, Joel David, Yao, Bokai |
|---|---|
| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $m\cap\kappa\in\kappa$. Comment: 36 pages, 6 figures. Commentary can be made on the first author's blog at http://jdh.hamkins.org/second-order-reflection-with-abundant-urelements. V2 contains several refinements, improvements to exposition, and additional citations |
| Databáze: | arXiv |
| Externí odkaz: |
|