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.
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