Difference between revisions of "Axiom of Choice"

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

Revision as of 09:16, 13 April 2007

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

Or more compactly

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.

Despite its usefulness, many mathematicians nowadays reject the axiom of choice. This rejection is based on the belief that all mathematical proofs should be constructive. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. The formulation of constructive axiom of choice is one of three major problems which challenge 21st century logicians.

The axiom of choice is not related to the liberal term pro-choice, which is the term used for themselves by proponents of abortion.