#MathsInMinutes Day 32: The axiom of choice
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The axiom of choice is a fundamental rule that is often added to the list of axioms used to define mathematical thinking. It is used implicitly in Cantor´s diagonal argument, and many other mathematical proofs that involve assuming infinite lists have some abstract existence, and that an infinite set of choice can be made.
More precisely, these proofs state that, given an infinite number of non-empty sets containing more than one element, it is possible to choose an infinite sequence of elements with precisely one from each set. To some this seems absurd – infinity rearing its awkward head again – but the rule allowing such a procedure is the axiom of choice.
Other axioms can be chosen, which allow the axiom of choice to arise as a theorem, but whichever version is used, this addition to the basic set of logical rules is necessary to make such arguments permissible.
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