Borel canonization of analytic sets with Borel sections
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Abstract:
Kanovei, Sabok and Zapletal asked whether every proper $\sigma$-ideal satisfies the following property: given $E$ an analytic equivalence relation with Borel classes, there exists a set $B$ which is Borel and $I$-positive such that $E\restriction _{B}$ is Borel. We propose a related problem – does every proper $\sigma$-ideal satisfy: given $A$ an analytic subset of the plane with Borel sections, there exists a set $B$ which is Borel and $I$-positive such that $A\cap (B\times \omega ^{\omega })$ is Borel. We answer positively when a measurable cardinal exists, and negatively in $L$, where no proper $\sigma$ ideal has that property. We show that a positive answer for all ccc $\sigma$-ideals implies that $\omega _{1}$ is inaccessible to the reals and Mahlo in $L$.References
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Additional Information
- Ohad Drucker
- Affiliation: Einstein Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram. Jerusalem, 9190401, Israel
- Received by editor(s): January 4, 2017
- Received by editor(s) in revised form: May 7, 2017, and May 31, 2017
- Published electronically: March 30, 2018
- Additional Notes: This paper is part of the author’s PhD thesis written at the Hebrew University of Jerusalem under the supervision of Professor Menachem Magidor.
- Communicated by: Mirna Džamonja
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3073-3084
- MSC (2010): Primary 03E15, 03E35, 03E55, 28A05, 54H05
- DOI: https://doi.org/10.1090/proc/13837
- MathSciNet review: 3787368