The hierarchy of omega1-Borel sets

2011/07/25 by Arnold W. Miller, Miller, Arnold W.
Mathematics · #03E15 #03E35 #03E50 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03E15 #msc:03E35 #msc:03E50

paper · pdf · doi:10.48550/arxiv.1107.5022

LaTex2e: 12 pages Latest version at: http://www.math.wisc.edu/~miller

arxiv created 2011/07/25 · arxiv updated 2011/07/26

Abstract

The family omega1-Borel sets is the smallest family of subsets of the real line which contains the family of open sets and is closed under complementation and omega1 unions. We show: Theorem 1. MA+notCH implies this hierarchy has length omega2. Theorem 2. In the Cohen real model it has length either omega1+1 or omega1+2.

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• Long Borel Hierarchies
• Generalized Borel Sets
• Combinatorial principles from adding Cohen reals
• Countable ranks at the first and second projective levels
• Descriptive set theory and forcing; How to prove theorems about Borel sets the hard way