Abstract

We show that several theorems about Polish spaces, which depend on the axiom of choice ), have interesting corollaries that are theorems of the theory \, where \ is the axiom of dependent choices. Surprisingly it is natural to use the full \ to prove the existence of these proofs; in fact we do not even know the proofs in \. Let \ denote the axiom of determinacy. We show also, in the theory \\), a theorem which strenghtens and generalizes the theorem of Drinfeld and Margulis about the unicity of Lebesgue’s measure. This generalization is false in \, but assuming the existence of large enough cardinals it is true in the model \,\in \rangle \).