Independence, Randomness and the Axiom of Choice

Journal of Symbolic Logic 57 (4):1274 - 1304 (1992)
Abstract
We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.
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