For the author to say "The Axiom of Choice is Wrong" is simply shorthand for:

    We make choices as to what we use as axioms in
    mathematics.  Having made the choices, we then
    work to see what consequences we are forced to
    accept.  In the case of the Axiom of Choice, I
    find some of the consequences unacceptable. As
    a result I decide to believe that the Axiom of
    Choice is not true.

I think it's more arbitrary than that, quoting his comment:

> Why do I believe these things? For no better reason than the crude generalized-intuition arguments like above. However, I also think its important to remember how baseless these beliefs are. Its like how I can root for a football team, but still remember that my allegiance is due primarily to proximity and nothing more meaningful.

I think that on average people who deny the axiom of choice are the people who approach set theory and probability theory with intuition, and then rather than discard their intuition as being incorrect, they discard the axiom of choice as not lining up with their intuition (and then go on to do great math in other fields). But seeing as the author is an algebraic geometer, and published this post early in graduate school, I'm willing to bet he switched his ideology. I don't think AoC is something you can deny while actively working in his field, since you need to make tons of assumptions about algebraic closures to even get started.

So if you are willing to go full Grothendieck from the get-go you can dodge the need to have algebraic closures. Even making an algebraic closure doesn't require the axiom of choice, as you can adjoin the roots of all polynomials with coefficients in your field without invoking it.

The problem is that a lot of categorical constructions involve sets that get big. This isn't too much of a problem, but if you want to show that they are nonempty, choice comes in in a big way. Even worse, ZFC isn't enough: you need to play tricks to formalize category theory inside of set theory (or use proper classes instead).

If a mathematician would throw away his mathematical understandings in order to prove more results, I worry for his soul. A theorem in ZFC-but-not-ZF is much less interesting than a theorem in ZF, except as a topic in model theory.

Mathematical understanding quite often contradicts intuition. Part of becoming a mathematician is learning when to trust your intuition, and when you start your mathematical career your intuition is pretty unreliable. The thing about AoC and measure theory is that it's notorious for challenging your intuition after you think you've built up a lot of good intuition.

Also, from my experience the vast majority of mathematicians don't care whether their theorems are ZF-compliant or need ZFC, unless it's marketed to logicians or specifically deals with AoC in a certain field.

Since you've edited your comment, I've deleted my reply, and here's a replacement ...

  > That's not true. Set theory is based on a lot of
  > assumptions and it only consistent if you assume
  > all of them are true.

All of mathematics is based on the idea of taking some axioms, and then seeing what follows. We usually use axioms that somehow align with our intuition or experience, and then see if things develop as we expect. Consistency is nothing to do with "Truth".

  > I could argue if you start at infinity and count
  > down you never reach zero as there are uncountable
  > many infinite numbers.

This doesn't make sense. To "start at infinity," what does it mean to count down at all? Certainly I can't interpret what you are saying in any meaningful way. Perhaps you could restate it in a more precise way.

  > Worse there are uncountable many digits in an
  > infinite number ...

There is no such thing as "an infinite number."

  > ... so any operation based on an infinite sequence
  > never finishes.

With care, operations on infinite sequences can be defined precisely and shown to be well-defined, so it's not clear what you're trying to say here.

  > Making that argument and much of set theory
  > meaningless.

??

  > PS: There are many Axioms which are chosen simply
  > because it makes math more interesting. One of the
  > more fun ones is to do set theory but allow for
  > some undefined relationships.

So you are creating a completely different math? Good, but not relevant here.

  > Aka ...

I don't think you mean "aka" - that means "Also known as ...". I suspect you mean something more like "For example" which is "E.g."

  > ... x is in set A but it's undefined if it's in
  > set B. Now quick draw a Venn diagram.

You haven't defined any of your terms, and you're doing something completely different from standard set theory. Sure, it might be fun, but it's more likely a toy problem in a toy theory. Can you give more precise definitions for what you're doing? Perhaps you should write it up and submit a link.

The comment section on this blog post is spectacular. Many of the comments are by very accomplished mathematicians.