>>11073122>>11074003>why it's remotely reasonableWe're stuck with AoC just for historical reasons. It has momentum in functional analysis and other frameworks and is thus hard to get rid.
When Cantor was young, in the 1870's or so, he worked on analysis and invented transfinite ordinals (N extended to w's) to make statements about compositions of trigonometric functions. This lead him to define set theory.
Around the same time, Frege came up with his predicate logic (quantifiers in a fully formal logic) and in the next 30 years, people would realize they can squeeze all of math into a formal set theory.
Then they realized that taking over intuitions from sets of naturals to a general notion of "set" is sketchy. For decades to come, people tried to wrap their head around the various notions of impredicativity. Some could be removed, some we still use today, some are not present in constructive set theory.
Here's a soft looking but mean example. If are resp. variables standing in for some natural number and som subset of natural numbers, consider the selection of natural numbers defined by
That's allowed to think about but already problematic. To inspect what the elements of T are, you must go through all subsets S of N, but T is among those, so you get in a cycle. But that was a soft digression to highlight the kind of issues..
The Axiom of Choice was written down by Zermelo for practical purposes around 1900 for some desirable ordering properties of sets, but it wasn't the focus of anyone back then.
ZF fixed the fatal impredicativity issues in set theory (Russel issues) by Fraenkels Axiom of Replacement, around 1920 and that's the one making the set theory universe V what is it (a transfinite sequence of applying the power set to N).
When Gödel explored his constructive ordering L inside V in 1940, Choice was already 40 years old and the standard was set.
