Whether a set is finite or infinite, its power set is always bigger.
Let S be any set and assume f: S -> P(S) exhaustively enumerates the subsets of S using only elements of S.
Then define a new subset L of S: For each s in S, we decide whether to include s in L as follows: If f(s) contains s, then L shall NOT contain s, if f(s) does not contain s, then L DOES contain it.
No s in S can be a preimage of L under f, since L disagrees with f(s) about whether it contains s or not, by construction. So f must have missed L. So P(S) is strictly larger than S.
Nothing about infinities in there. You can't have a surjective mapping from any set to its power set, ever.
Let S be any set and assume f: S -> P(S) exhaustively enumerates the subsets of S using only elements of S.
Then define a new subset L of S: For each s in S, we decide whether to include s in L as follows: If f(s) contains s, then L shall NOT contain s, if f(s) does not contain s, then L DOES contain it.
No s in S can be a preimage of L under f, since L disagrees with f(s) about whether it contains s or not, by construction. So f must have missed L. So P(S) is strictly larger than S.
Nothing about infinities in there. You can't have a surjective mapping from any set to its power set, ever.