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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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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