Changes

Or more compactly

Or more compactly

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

<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 Axiom of Choice has many equivalent statements, such as "the [[Cartesian product]] of any collection of nonempty sets is nonempty", [[The Well-Ordering Principle]], the existence of [[cardinal numbers]], the [[continuum hypothesis]], and the existence of subsets of the real line which do not have a well-defined [[Lebesgue measure]].

The Axiom of Choice has many equivalent statements, such as "the [[Cartesian product]] of any collection of nonempty sets is nonempty", [[The Well-Ordering Principle]], the existence of [[cardinal numbers]], the [[continuum hypothesis]], and the existence of subsets of the real line which do not have a well-defined [[Lebesgue measure]].