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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I wouldn’t say it’s really “controversial” nowadays. In the late 19th and early 20th centuries, mathematicians were wrestling with how to put the entirety of their field on a solid foundation, to try and settle some doubts about what mathematical techniques and arguments are actually valid. They kept coming up with exciting steps forward – for example, they came up with set theory and realised that you could describe pretty much every mathematical concept purely in terms of sets. But they also kept running into problems – for example, some early formulations of set theory turned out to be self-contradictory. The axiom of choice is a rule that you can include in a set theory, and it makes some things easier but also has some weird consequences. In those days, mathematicians were having lots of arguments about what rules make sense for set theories and how they can be justified, and the axiom of choice was one of the big controversies.
Nowadays, the consequences of AC are much better understood, and it has been realised that some of the properties that early 20th century mathematicians wanted their foundation of mathematics to have are not actually possible. Most mathematicians use the axiom of choice without really thinking about it, but some study systems in which it is not assumed (or is outright false), and some continue to study its consequences and alternative variants of it and so on. Very few people have particularly strong views about it.
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.
First of all, the axiom le choices fundamentally deals with infinity.
In other words, it deals with mathematical constructs that don’t have a clearly testable experiment like “take 2 objects plus 2 objects and check that there is 4 objects”.
The observervable universe is finite in size so it doesn’t matter if the actual universe is infinite. Limits to precision of measurements means that it doesn’t matter either if “infinitely small” is a thing.
That doesn’t mean that infinity is never used in physics. It’s used everywhere to model how physics works. But it’s a mathematical, and as all mathematical constructions there are multiple possibilities with different properties.
You can build mathematics with an infinity satisfying the axiom of choice. Or you can build mathematics with an infinity that doesn’t.
Which one you select makes no practical difference as soon as you fully enter the “application” side, but one might make the mathematics easier or harder depending on the circumstances.
So peoples use it when it’s practical, and forbid themselves from using it in the rare cases where that would make things harder.
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