Some theories
Axiom of choice: For an arbitrary collection of sets $\{A_{\alpha}\}_{\alpha\in J}$, there exists a subset $C$ of the disjoint union of the $A_i$s such that, $C\cap A_i$ is a singleton for $i\in J$.
Zorn's Lemma: For any non-empty partially ordered set $A$, for any totally ordered subset, if there exists an upper bound in $A$, then $A$ has a maximal element.
Definition: An ordered set is well ordered if we have a minimum element for every subset of $A$.
Well Ordering Principle: If $A$ is any set, then an ordered relation exists, which is well-ordering.
Theorem: The Axiom of choice, Zorn's Lemma, and the Well Ordering principle are all equivalent.
Definition: We define section of a set $A$, as $S_{\alpha}=\{x|x<\alpha,x\in A\}$
Lemma: There exists a well-ordered set A with a maximum element $\Omega$, such that $S_{\Omega}$ is uncountable but all other sections are countable.
In other words, $S_{\Omega}$ is an uncountable set whose sections are countable. It is denoted as the "minimal uncountable well-ordered set".
Proposition: If $B$ is a countable subset of $S_{\Omega}$, then it has an upper-bound in $S_{\Omega}$.
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