[openasocket]

Can you elaborate? It all seems really straightforward to me. There is no bijection between a set and its power set, via diagonalization. Thus, there is no bijection from the natural numbers to the power set of natural numbers. By definition, that means the power set of natural numbers is uncountable.

[smokel]

Not the parent, but the argument uses quite a few assumptions (axioms) that may not be intuitive to everyone, but which are quite relevant when studying mathematics at the foundational level.

For example, why would one be able to create the diagonal set (those indices of the power set elements that do not contain that index as an element) and the enumeration of the power set (i.e. the entire list of possible sets of numbers) at the same time? The theorem proves that an enumeration of the power set cannot be made. Perhaps some sets cannot be constructed at will just by writing down its properties either?

In computer land, one would quickly run into self-referential problems when constructing sets like these. For mathematics of this kind, most people agree that this is all fine, and one can derive interesting things from it. But one can also reject the approach and still do some elementary fun stuff.

Then again, I might be completely misunderstanding all of this, and I love to be corrected.

Edit: wording

[openasocket]

I’m not sure there’s many axioms used. Given any set A, and a function from A to the power set, P(A), construct the set X = {a in A | a is not in f(a) }. Here all we’re using is the power set axiom to define the power set and the subset axiom schema to construct X. We claim there is no a such that f(a) = X. If there was such an a, is a in X? By construction, a is in X if and only if a is not in X, just by first order logic, which is a contradiction. Thus, X is not in the image of f, so f is not a bijection. Thus, there is no bijection from A to P(A). And that’s it. We don’t even need the axiom of choice

[cubefox]

You can't prove such a thing in ZFC [1]. ZFC's "power set axiom" is a misnomer. It doesn't imply the existence of infinite power sets. It just says if any set S exists and is a subset of an infinite set A, then S is element of a set P (the supposed "power set"). But the axiom doesn't imply the existence of "all" subsets of A. There is no way for first-order logic to state "every possible combination of elements of A forms a set". And Cantor's theorem stating that there is no mapping f from A onto P merely means that the mapping f itself can't exist inside a model of ZFC. [2]

[1] Or any other theory of first-order logic.

[2] This is explained in more detail in Stewart Shapiro's book "Foundation without Foundationalism" p. 114f.

[4death4]

> But the axiom doesn't imply the existence of "all" subsets of A.

I feel like that would be a consequence of the axiom of choice.

[cubefox]

At the moment I don't remember enough of the axiom of choice to comment on it in detail, but we can infer from the Downward Löwenheim-Skolem theorem (roughly, every first-order theory, including ZFC, has a countable model) that the Choice axiom can't imply uncountable sets.

[4death4]

Admittedly I know nothing about the theorem you’re referring to, but is there a compelling reason to limit ourselves to first-order logic?

[openasocket]

I’m uncertain what your point is. Like I said, in ZFC, there is no bijection from A to its power set. I don’t think anything you’ve stated contradicts that. Are there alternative axiom systems in which you can construct such a bijection?

[cubefox]

There is no bijection (surjection, to be more general) from A to P inside your ZFC model even if both A and P are assumed to be countable. The bijection we talk about here would be just another object inside the model. So you can't infer from the lack of such an object that P is uncountable.

[openasocket]

Yeah, Cantors theorem is a theorem written in the language of ZFC set theory. In other axiom systems it may not be true. But you can also say that about literally every theorem beyond, like, modus ponens. Is that the point you are trying to make?