Probability Axioms and Set Theory Paradoxes

In this paper, we show that Zermelo-Fraenkel set theory with Choice (ZFC) conflicts with basic intuitions about randomness. Our background assumptions are the Zermelo-Fraenekel axioms without Choice (ZF) together with a fragment of Kolmogorov's probability theory. Using these minimal assumptions, we prove that a weak form of Choice contradicts two common sense assumptions about probability-both based on simple notions of symmetry and independence.