The axiom of choice

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The axiom of choice is an axiom, usually adjoined to the Zermelo-Frenkael axioms of set theory. Informally the axiom of choice says that given a family of non-empty sets we have a function selecting exactly one element from each set. The axiom is equivalent to the assertion that every set can be well ordered, and equivalently that all the infinite cardinals are $\aleph$-numbers. The axiom has many equivalents, including but not limited to:

  • The well-ordering principle, which states that every set can be well-ordered.
  • Zorn's lemma, stating that if the union of any chain in $x$ is itself a member of $x$, then $x$ must have some maximal element, with respect to inclusion.
  • Cantor's Law of Trichotomy, the statement that any two sets are comparable with respect to cardinality, that is, $|A|\lt|B| \lor |A|=|B| \lor |B|\lt|A|$


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