In $ZF$, $AC$ is equivalent to $\forall \alpha (\mathscr P(\alpha)$ can be well-ordered)
42314
Without appealing to choice, can we prove that if $X$ is well-orderable, then so too is $2^X$?
150605
Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$
149277
Does well-ordering of the proper class of cardinal numbers imply choice?
113863
The non-existence of non-principal ultrafilters in ZF
43024
Well-ordered sets, properties of an element that is not largest in the set
145752
Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
35972
Well ordering principle for rationals
13162
Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit
78561
Algebraic field extensions: Why $k(\alpha)=k[\alpha]$.
108957
Cardinality of a totally ordered union
113065
Proving equivalences of statements equivalent to AC
56370
Associative operation on a set $S$
72371
Proof of Strong Induction Using Well-Ordering Principle
120213
Example $x$, $y$ and $z$ values for $x\uparrow^\alpha y=z$ where $\alpha\in \Bbb R-\Bbb N$
41505
Transcendental Extensions. $F(\alpha)$ isomorphic to $F(x)$
55978
Prove that for every countable ordinal $\alpha$, $\alpha\times[0,1)$ is order isomoprhic to $[0,1)$
110654
Vector Spaces and AC