{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:44:52Z","timestamp":1760233492666,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T00:00:00Z","timestamp":1611273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we show that Zermelo\u2013Fraenkel set theory with Choice (ZFC) conflicts with basic intuitions about randomness. Our background assumptions are the Zermelo\u2013Fraenekel axioms without Choice (ZF) together with a fragment of Kolmogorov\u2019s probability theory. Using these minimal assumptions, we prove that a weak form of Choice contradicts two common sense assumptions about probability\u2014both based on simple notions of symmetry and independence.<\/jats:p>","DOI":"10.3390\/sym13020179","type":"journal-article","created":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T12:28:31Z","timestamp":1611577711000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Probability Axioms and Set Theory Paradoxes"],"prefix":"10.3390","volume":"13","author":[{"given":"Ari","family":"Herman","sequence":"first","affiliation":[{"name":"Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97201, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2692-9198","authenticated-orcid":false,"given":"John","family":"Caughman","sequence":"additional","affiliation":[{"name":"Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97201, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,22]]},"reference":[{"key":"ref_1","first-page":"1136","article-title":"Problem 5348","volume":"72","author":"Galvin","year":"1965","journal-title":"Am. 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