Journal
SYMMETRY-BASEL
Volume 13, Issue 2, Pages -Publisher
MDPI
DOI: 10.3390/sym13020179
Keywords
set theory; probability; axiom of choice
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The paper demonstrates that Zermelo-Fraenkel set theory with Choice conflicts with basic intuitions about randomness, showing contradiction between a weak form of Choice and common sense assumptions about probability based on symmetry and independence.
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.
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