A dedekind finite borel set

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 is a G δσ -set then either B is countable or B contains a perfect subset. Second, we prove that if 2 ω is the countable union of countable sets, then there e...

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Veröffentlicht in:	Archive for mathematical logic 2011-02, Vol.50 (1-2), p.1-17
1. Verfasser:	Miller, Arnold W.
Format:	Artikel
Sprache:	eng
Schlagworte:	Algebra Archives Axioms Borel sets Construction Mathematical analysis Mathematical logic Mathematical Logic and Foundations Mathematical models Mathematics Mathematics and Statistics Proof theory Theorems Theoretical mathematics Unions
Online-Zugang:	Volltext
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Zusammenfassung:	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 is a G δσ -set then either B is countable or B contains a perfect subset. Second, we prove that if 2 ω is the countable union of countable sets, then there exists an F σδ set such that C is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite which is F σδ .
ISSN:	0933-5846 1432-0665
DOI:	10.1007/s00153-010-0195-6