2008/06/11 by Arnold W. Miller, Miller, Arnold W.
Computer Science · Mathematics · #03E15 #03E25 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO #msc:03E15 #msc:03E25
paper · pdf · doi:10.48550/arxiv.0806.1957
Latex2e: 21 pages Latest version at http://www.math.wisc.edu/~miller
arxiv created 2008/06/11 · openalex publication_date 2008/06/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we prove that if the real line is the countable union of countable sets, then there exists an F-sigma-delta set which is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite set of reals which is F-sigma-delta.