The law of infinite cardinal addition is weaker than the axiom of choice
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- by J. D. Halpern and Paul E. Howard PDF
- Trans. Amer. Math. Soc. 220 (1976), 195-204 Request permission
Abstract:
We construct a permutation model of set theory with urelements in which ${C_2}$ (the choice principle restricted to families whose elements are unordered pairs) is false but the principle, “For every infinite cardinal m, $2m = m$” is true. This answers in the negative a question of Tarski posed in 1924. We note in passing that the choice principle restricted to well-ordered families of finite sets is also true in the model.References
- J. D. Halpern and Paul E. Howard, Cardinal addition and the axiom of choice, Bull. Amer. Math. Soc. 80 (1974), 584–586. MR 329890, DOI 10.1090/S0002-9904-1974-13510-X
- Ernst Specker, Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom), Z. Math. Logik Grundlagen Math. 3 (1957), 173–210 (German). MR 99297, DOI 10.1002/malq.19570031302 Alfred Tajtelbaum-Tarski, Sur quelques théorèmes qui équivalent à l’axiome du choix, Fund. Math. 5 (1924), 147-154. —, Problème 31, Fund. Math. 5 (1924), 338.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 220 (1976), 195-204
- MSC: Primary 02K20; Secondary 02K05
- DOI: https://doi.org/10.1090/S0002-9947-1976-0409183-1
- MathSciNet review: 0409183