Changes

The Axiom of Choice is an [[axiom]] of ZFC set theory that states the following:

The Axiom of Choice is an [[axiom]] of ZFC set theory that states the following:

<math>\forall x\;(\forall y\;y\in x\Rightarrow\exists z\; z\in y)\;\exists S\;(\forall z\;z\in x\Rightarrow\exists w\;w\in z\;\wedge\;w\in S)\;\wedge\;(\forall v\;v\in z\;\wedge\;v\in S\Rightarrow v=w)</math>

::<math>\forall x\;(\forall y\;y\in x\Rightarrow\exists z\; z\in y)\;\exists S\;(\forall z\;z\in x\Rightarrow\exists w\;w\in z\;\wedge\;w\in S)\;\wedge\;(\forall v\;v\in z\;\wedge\;v\in S\Rightarrow v=w)</math>

Or more compactly

Or more compactly

<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math>

::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math>

The English translation of the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We have the right to choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set."

The English translation of the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We have the right to choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set."