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.∎

单词	WellorderingPrincipleImpliesAxiomOfChoice
释义	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.∎
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