Does there exists a set X where the power set of X is countably infinite? I strongly feel the answer is no, however I can't provide a formal proof of why such a set cannot exist.
Existence of a Set with Countably Infinite Power Set?
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elementary-set-theory
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0It can be shown by eliminating the possibilities, e.g. can $X$ be finite? Can it be more than finite? – 2017-02-04
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0A finite set has a finite power set. A countably infinite set has a uncountably infinite power set. Anything larger than countable will have a much larger power set. – 2017-02-04