Axiom of dependent choice

In mathematics, the axiom of dependent choice is a weak form of the axiom of choice which is still sufficient to develop most of real analysis. The axiom can be stated as follows: For any nonempty set ''X'' and any entire binary relation ''R'' on ''X'', there is a sequence (''x''_''n'') in ''X'' such that ''x''_''n''''R''''x''_''n''+1 for each ''n'' in N. (Here an ''entire'' binary relation on ''X'' is one such that for each ''a'' in ''X'' there is a ''b'' in ''X'' such that ''aRb''.) Note that even without such an axiom we could form the first ''n'' terms of such a sequence, for any natural number ''n''; the axiom of dependent choice merely says that we can form a whole sequence this way, which is intuitively obvious. ''See also: axiom of countable choice'' Set theory

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Axiom_of_dependent_choice
Axiom_of_dependent_choices

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