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EQUIVALENCE RELATIONS WHICH ARE BOREL SOMEWHERE

Published online by Cambridge University Press:  08 September 2017

WILLIAM CHAN
WILLIAM CHAN*
Affiliation:
DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91106, USA E-mail: wcchan@caltech.edu
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Abstract

The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I + ${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$ . If for all $z \in {H_{{{\left( {{2^{{\aleph _0}}}} \right)}^ + }}}$ , z ♯ exists, then there exists an I + ${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence relation.

Keywords

03E1503E3503E55equivalence relationsproper forcingsidealsdescriptive set theoryBorel setsanalytic sets

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 893 - 930
Copyright
Copyright © The Association for Symbolic Logic 2017 

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