# Is axiom of choice applicationless?

Hurkyl
Staff Emeritus
Gold Member
Hrm, let me amend that... with the Cauchy sequence definition, I think you're almost right. There are exactly three subsets of R:
1. The empty set (everything is rejected)
2. R (everything is accepted)
3. The set whose membership relation is always "no answer".

However, other definitions for real numbers (e.g. nested sequence of intervals whose endpoints are rational and whose length goes to zero) would allow more interesting subsets of R.

I suspect that Zorn's lemma is required to find the basis for a general Sobolev space.

If this is true, you could look for examples of Sobolev spaces that require Zorn's lemma for a basis, and then invent a PDE using that space.
When a Sobolev space is used to prove an existence of a solution to some PDE, isn't it the completeness of that Sobolev space that is essential, and not any particular basis?

(I don't know much of Sobolev spaces, but some lecture notes of one other course mentioned something about them in the beginning, so I have some idea.)

Does there exist a theorem, whose proof relies on the axiom of choice, and which has practical or concrete applications? I'll consider for example PDE and number theory problems practical or concrete, because they can be intuitively connected with the physical world. On the other hand, for example non-measurable sets hardly have effect on anything, except on abstract theory.
Even if there was such a theorem, what makes you think you can chose it? 