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.
I know what it is, but how does it relate to 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.
It is equivalent to the axiom of choice.
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:
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.
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Every injective function has a left inverse and that doesn't require the axiom of choice and both feel equally intuitive to me
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