This post is the April fools post. As such, this post is strictly for satirical purposes

If anyone on the Fields Medal committee reads this, my contact information is at the bottom of this post 😎

S tier

Proof by human extinction

Consider the set of elements that have been chosen from a set thus far in human history. Call this set S. Because humans can only name a finite number of things during their lifetime, and the total number of human lives will be finite, S is finite. Therefore there are real numbers that are not included in S. Therefore there exists a set from which no function can pick an element. Therefore the axiom of choice is false.

A tier

Proof by bullying

I mean you just kind of pick one. C'mon give me a set, I'll be able to pick an element out of it. This isn't that hard. Why would you even think this is false? What are you, some kind of idiot?

Proof by utilitarianism

If someone were to prove the axiom of choice then they would surely become rich and famous. I vow to donate any money I make from the notoriety I would get from proving the axiom of choice to charity, saving potentially millions of lives. Math is a human construction, therefore it is up to us to decide if a proof is valid. By believing this proof to be valid you increase the average number of lives saved. Therefore you have a moral obligation to believe this proof to be valid.

B tier

Proof by experts

The majority of professional mathematicians assume the axiom of choice¹. Therefore, it is true. I mean they are the experts.

C tier

Proof by layer cake

Each set has its own set of choice functions. So for each set, just pick a choice function out of that set of choice functions, and use it.

Proof by tautological induction

Consider the set where the maximum size of any set in it is 1. Then the choice function is just the element in the set. Now suppose it is true for sets where the maximum size is n. It must also be true for size n+1 because you can just pick one element to exclude and pick out of the remaining n.

D tier

Deus ex machina

Can't we just plug each set into chat gpt and get an element out?