| > I don't see how you can be _skeptical_ of those ideas. Well you can be skeptical of anything and everything, and I would argue should be. Addressing your issue directly, the Axiom of Choice is actively debated: https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_... I understand the construction and the argument, but personally I find the argument of diagonalization should be criticized for using finities to prove statements about infinities. You must first accept that an infinity can have any enumeration before proving its enumerations lack the specified enumeration you have constructed. https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument > Math is math, if you start with ZFC axioms This always bothers me. "Math is math" speaks little to the "truth" of a statement. Math is less objective as much as it rigorously defines its subjectivities. https://news.ycombinator.com/item?id=44739315 |
The axiom of choice is not required to prove Cantor’s theorem, that any set has strictly smaller cardinality than its powerset.
Actually, I can recount the proof here: Suppose there is an injection f: Powerset(A) ↪ A from the powerset of a set A to the set A. Now consider the set S = {x ∈ A | ∃ s ⊆ A, f(s) = x and x ∉ s}, i.e. the subset of A that is both mapped to by f and not included in the set that maps to it. We know that f(S) ∉ S: suppose f(S) ∈ S, then we would have existence of an s ⊆ A such that f(s) = f(S) and f(S) ∉ s; by injectivity, of course s = S and therefore f(S) ∉ S, which contradicts our premise. However, we can now easily prove that there exists an s ⊆ A satisfying f(s) = f(S) and f(S) ∉ s (of course, by setting s = S), thereby showing that f(S) ∈ S, a contradiction.