Every set can be well-ordered and you cannot convince me otherwise. AOC good.
I think the main reason is she seems supportive of US Imperialism. When the government does stuff redistribute the spoils of imperialism socialism is less than ideal to be sure.
Who is "she"? I don't see what imperialism has to do with age of consent.
I had assumed AOC mean Alexandria Occasio Cortez, a US American liberal politician.
USA? Liberals? Comrade did you hit your head? Those haven't existed for 30 years.
Oh, so I bet you think that you can make two spheres out of one that are the same size as the original, too. The liberal has a peculiar kind of madness. Pure ideology. :zizek-ok:
Every set can be well-ordered
what does this have to do with AOC?
The Axiom of Choice (AOC) is necessary to construct a Zermelo-Fraenkel set theory that admits the well-ordering property. This version of set theory, ZFC, is the version that we use for everyday math because it has a lot of properties that seem natural. As the post implies, there is actually a bit of controversy here. The axiom of choice implies the existence of some (very pathological and complicated) sets that you can also prove are impossible to construct. Mathematicians tend to be bothered by things that not only exist, but you can prove you are unable to meaningfully examine.
I hate the list of equivalent statements. Some are completely intuitive, others seem outlandish, but none seem like they should be related to eachother at all.
Honestly I'm pretty comfortable with most of the equivalent statements. Here are the statements I find especially intuitive:
- The axiom of choice itself (the cartesian product of an infinite number of sets is non-empty)
- Every surjective function has a right inverse
- Trichotomy of set cardinality
- Zorn's Lemma/Every vector space has a basis/Every ring has a maximal ideal/Every group has a maximal subgroup/Every connected graph has a spanning tree
- Tychonoff's theorem
- This one is equivalent to countable choice, but that a countable union of countable sets is countable
- Well-ordering theorem
Some of the weirder statements implied by it like Banach-Tarski/existence of non-Lebesgue measurable sets I just personally chalk up to general infinity weirdness in the same vein as something like Hilbert's Hotel.
Every injective function has a left inverse and that doesn't require the axiom of choice and both feel equally intuitive to me
This is the best take on the site. We've done it, folks. Shut this place down.
