Login
This is a place holder for potential sponsor logos.
well-ordering principle implies axiom of choice
Theorem The well-ordering principle implies the axiom of choice.
Proof. Let $C$ be a collection of nonempty sets. Then $\displaystyle \bigcup_{S \in C} S$ is a set. By the well-ordering principle, $\displaystyle \bigcup_{S \in C} S$ is well-ordered under some relation $<$ . Since each $S$ is a nonempty subset of $\displaystyle \bigcup_{S \in C} S$ , each $S$ has a least member $m_S$ with respect to the relation $<$ .
Define $\displaystyle f \colon C \to \bigcup_{S \in C} S$ by $f(S)=m_S$ . Then $f$ is a choice function. Hence, the axiom of choice holds. 
well-ordering principle implies axiom of choice is owned by Warren Buck.
None.