High Quality Content by WIKIPEDIA articles! Transfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. Let P(?) be a property defined for all ordinals ?. Suppose that whenever P(?) is true for all ? < ?, then P(?) is also true. Then transfinite induction tells us that P is true for all ordinals. That is, if P(?) is true whenever P(?) is true for all ? < ?, then P(?) is true for all ?. Or, more practically: in order to prove a property P for all ordinals ?, one can assume that it is already known for all smaller ? < ?.

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