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.
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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