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 ${B\subseteq 2^\omega}$ is a G _??? -set then either B is countable or B contains a perfect subset. Second, we prove that if 2 ^?? is the countable union of countable sets, then there exists an F _??? 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 _??? .