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Aspects of Definability for Equivalence Relations


Chan, William (2017) Aspects of Definability for Equivalence Relations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z90P0X3M.


This thesis will show that in the constructible universe L and set forcing extensions of L, there are no almost Borel reductions of the well-ordering equivalence relation into the admissibility equivalence relation and no Borel reductions of the isomorphism relation of any counterexample to Vaught's conjecture into the admissibility equivalence relation.

Let E be an analytic equivalence relation on a Polish space X with all classes Borel. Let I be a sigma-ideal on X such that its associated forcing of I-positive Borel subsets is a proper forcing. Assuming sharps of appropriate sets exist, it will be shown that there is an I-positive Borel subset of X on which the restriction of E is a Borel equivalence relation.

Assuming there are infinitely many Woodin cardinals below a measurable cardinal, then for any equivalence relation E in L(R) with all Borel classes and sigma-ideal I whose associated forcing is proper, there is an I-positive Borel set on which the restriction of E is Borel.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Descriptive Set Theory, Equivalence Relation, Admissibility, Constructibility, Forcing, Large Cardinals
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kechris, Alexander S.
Thesis Committee:
  • Kechris, Alexander S. (chair)
  • Graber, Thomas B.
  • Flach, Matthias
  • Lupini, Martino
Defense Date:17 May 2017
Funding AgencyGrant Number
National Science FoundationDMS 1464475
National Science FoundationEMSW21-RTG DMS 1044448
Record Number:CaltechTHESIS:05312017-155530848
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for chapter 2.
Chan, William0000-0002-0661-1764
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10236
Deposited By: William Chan
Deposited On:02 Jun 2017 20:06
Last Modified:04 Oct 2019 00:16

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