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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The law of infinite cardinal addition is weaker than the axiom of choice


Authors: J. D. Halpern and Paul E. Howard
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
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0409183-1
Article copyright: © Copyright 1976 American Mathematical Society

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