Why is the Axiom of Choice kind-of controversial? People on the internet are talking whether it’s actually needed or something and I don’t get it 🙁
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Math is just a set of rules – especially types of math like set theory, especially when we’re dealing with things like infinity. Our initial instinct is to think of mathematical problems as happening in the real world, but that kind of breaks down when we’re dealing with more abstract mathematics.
To do different kinds of math, we choose different rules. It’s interesting to see what kinds of results you get with different kinds of rules. One of the best examples of this is geometry – one of the rules Euclid worked with is “if two lines are parallel, they will never intersect.” Turns out, if you take away this rule, you can do all kinds of interesting math and get important results that do have real-world applications.
The axiom of choice is one of those rules in the field of set theory, a rule that doesn’t really have a real-world parallel. It’s a rule about choosing an infinite number of things, which is something we can’t do in the real world.
If you don’t include it, there are a whole lot of questions that are unanswerable – that is, if the question is “can we do this?”, the answer is “we can’t prove that we can, but we **also** can’t prove that we **can’t**”. So, in short, the mathematics is in some sense less “useful” or less interesting.
If you *do* include it, a lot of questions become solvable. But you also get results that don’t match how we view the real world. The math becomes more powerful perhaps, but in some sense it also becomes less intuitive/comfortable.
So, the question of whether to include the axiom of choice in the rules you’re working with is largely one of need and philosophy. Plus, if you have a question you don’t know the answer to, you can get useful information from figuring out the related questions: “do I need the axiom of choice to be true (or false) to solve this question? Can I solve this question regardless of whether it’s true? Will I get one answer to my question if the axiom of choice is true, and an opposite answer if it’s false?”
That’s the kind of thing people are thinking about when they talk about whether the axiom of choice is necessary.
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