mattisbusiness wrote:The reality is that we don't really deal with infinite things in the real world, so its consequences don't really matter.
I've probably never disagreed more with something that I've read on these boards. All of calculus is based on the idea of infinities, so do the consequences of calculus not really matter?
You can do calculus without infinities.
Using infinities can make it easier and less messy.
I work in quantum computing, and here are two things that the Axiom of Choice does just in that one particular field that I can think of off the top of my head:
- AoC implies the Hahn-Banach theorem, which in turn implies Arveson's Extension theorem, which says that any completely positive map defined on an operator system can be extended to a completely positive map on the entire overlying space. What this does in quantum computing is ensures that if you have a quantum channel defined on an algebra (relatively nice small subsection of you whole space), you can twerk it to make sure that it also behaves nicely elsewhere.
You read it wrong. If you can actually twerk it so that it behaves nicely elsewhere, then you didn't need to AoC to prove it for that case
. The cases for which the AoC would be needed to prove that it "could be twerked" are those that you cannot actually find the twerking for.
Basically, you can do all of your work in ZF~C instead of ZFC, and anything that applies to concrete reality still holds -- because the AoC is about infinite cross products of sets, not about finite cross products of sets.
Now, AoC helps -- you can presume the basis without finding it, or presume there exists an exact solution that you are finding successive approximations to, instead of having to prove that there is a means of producing successive answers that, in some sense, approximate the actual answer. Lots of wording gets better.
The vector spaces for which you can find an explicit basis for - they have a basis regardless of AoC. The vector spaces that you cannot
find the basis for, those are the ones that AoC claims that there is a basis for them.
Having the AoC around helps lots. But don't give it too much credit -- much of mathematics can be done without it, it is just much