If $S$ is a non-empty set, then we can take $r$ such that $r \in S$. Why do we need the axiom of choice?
48030
Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.
153935
A weaker Axiom of Infinity?
12539
Set Theory: Equivalency of Axiom of Choice and Choice Function
40291
What's the point of the axiom of extensionality?
33563
Is the axiom of choice used in the proof that every open set is the union of basis elements (Mukres, Lemma 13.1)
62936
Proving equivalencies of two versions of the Axiom of Choice
120701
Why do we need to learn Set Theory?
88823
How can the empty set be open in the definition of the cofinite topology?
120064
Proving that without the axiom of choice there is a set with an accumulation point that isn'tthe limit of a convergeant sequence
106670
Existence of vector space complement and axiom of choice
47697
Russell's paradox: a set cannot contain its own powerset
146188
Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice?
40232
What do we mean when we say "Let $x$ be an element of the set $\mathbb{R}$"?
82169
Why does a partially ordered set need to be reflexive?
31212
The existence of the empty set is an axiom of ZFC or not?
81509
The cardinality of a countable union of countable sets, without the axiom of choice
84378
Does axiom of foundation/regularity protect against Russell-like paradoxes?