2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162762In 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.Latex2e: 21 pages Latest version at http://www.math.wisc.edu/~millerLogic03E25, 03E15text