Can we always produce a bijective from from an uncountable set $A$ to an uncountable set $B$ ? Assume $A, B \subset \Bbb R$
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4See this. – David Mitra Feb 03 '14 at 15:00
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3If $A$ and $B$ are both Borel sets, then YES. – GEdgar Feb 03 '14 at 15:13
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See this possible duplicate: http://math.stackexchange.com/questions/650654/set-in-between-integers-and-reals/ – Asaf Karagila Feb 03 '14 at 16:05
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Whether there is any subset of $\mathbb{R}$ strictly smaller than $\mathbb{R}$ and strictly greater than $\mathbb{N}$ is not provable in the standard set theory, even with the axiom of choice. So the answer is either "no" (in general) or "it depends" (on the chosen model).
rewritten
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If you by "produce a bijective" means actually writing it down, constructively, then the answer must certailnly be NO.
kjetil b halvorsen
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