Archive for Mathematical Logic 50 (1-2):1-17 (2011)

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 2 ω 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
Keywords Dedekind finite  Perfect set  Countable sets of reals  Countable axiom of choice
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DOI 10.1007/s00153-010-0195-6
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Believing the Axioms. I.Penelope Maddy - 1988 - Journal of Symbolic Logic 53 (2):481-511.
The Axiom of Choice.Thomas J. Jech - 1973 - Amsterdam, Netherlands: North-Holland.
The Structure of Amorphous Sets.J. K. Truss - 1995 - Annals of Pure and Applied Logic 73 (2):191-233.

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