In other words I am looking for a set that has countably infinite number of subsets.
I have a feeling it doesn't exist, but if so, how do I prove it?
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Formal proof that $\mathbb{N}$ is the smallest infinite set + Cardinality of a set A is strictly less than the cardinality of the power set of A – Winther May 25 '18 at 13:16
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There is not such a set.
If its cardinal is finite then his power set is also finite. And if its cardinal is infinite, therefore at least countable, its power set has at least the continuum for cardinal.
mathcounterexamples.net
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And this is true without assuming choice, i.e., without assuming that an infinite set is at least countable. – Asaf Karagila May 25 '18 at 16:24