Axiom of Choice

Overview

This axiom assumes the existence of some choice function capable of selecting some element from a nonempty set. Note this axiom is controversial because it is non-constructive: there is no procedure we can follow to decide which element was chosen.

Equivalences

Relation Form

For any relation $R$ there exists a function $F\subseteq R$ with $\text{dom}F=\text{dom}R$.

Multiplicative Form

For any set $I$ and function $H$ with domain $I$, if $H(i)\ne \varnothing$ for all $i\in I$, then ${\mathbf{\times }}_{i\in I}H(i)\ne \varnothing$. Note this statement relies on the Cartesian product of infinite sets.

Covering Form

For any set $A$, there exists a function $F$ with $\text{dom}(F)=\mathcal{P}(A)-\{\varnothing \}$ such that $F(B)\in B$ for all $B\in \text{dom}(F)$.

Partition Form

Let $P$ be a partition of set $A$. Then there exists a set $B$ containing exactly one element from each member of $P$.

Cardinal Comparability

For any two cardinal numbers $\kappa$ and $\lambda$, either $\kappa \le \lambda$ or $\lambda \le \kappa$.

Zorn's Lemma

Let $A$ be a partially ordered set with ordering relation $\le$. A chain is a subset of $A$ that is totally ordered for the induced order. If every chain $B\subseteq A$ has an upper bound also in $A$, then Zorn's lemma states that $A$ must have a maximal element.

Arithmetic of Infinite Cardinals

Squaring

Assume the axiom of choice and let $\kappa$ be an infinite cardinal. Then $\kappa \cdot \kappa =\kappa$.

Absorption Law

Let $\kappa$ and $\lambda$ be cardinal numbers, the larger of which is infinite and the smaller of which is nonzero. Then