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A dedekind finite borel set

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Abstract

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 \({B\subseteq 2^\omega}\) is a G δσ -set then either B is countable or B contains a perfect subset. Second, we prove that if 2ω is the countable union of countable sets, then there exists an F σδ set \({C\subseteq 2^\omega}\) 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 \({D\subseteq 2^\omega}\) which is F σδ .

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  1. Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI, 53706-1388, USA

    Arnold W. Miller

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  1. Arnold W. Miller
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Correspondence to Arnold W. Miller.

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Miller, A.W. A dedekind finite borel set. Arch. Math. Logic 50, 1–17 (2011). https://doi.org/10.1007/s00153-010-0195-6

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  • DOI: https://doi.org/10.1007/s00153-010-0195-6

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