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Suppose $\bigcup_{n=1}^{\infty}A_n $ has cardinality continuum, prove that at least one $A_n$ has cardinality continuum.

If the choice axiom (C.A.) holds, König's theorem can be used to prove it; if the continuum hypothesis (C.H.) holds, then proof by contradiction can be used; if neither the choice axiom nor the continuum hypothesis is used, can the proposition still be proved?

Asaf Karagila
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  • Related? https://math.stackexchange.com/a/199419/42969 – Martin R Sep 23 '22 at 09:06
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    No. It is consistent with ZF that the real line is the union of countably many sets of strictly smaller cardinality. (This should be in standard sources like Jech's book "The Axiom of Choice".) – Andreas Blass Sep 23 '22 at 14:29

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