Transfinite induction

A proof principle that establishes a property for every ordinal by proving it at each stage from its validity at all earlier stages.

Transfinite induction principle: A theorem schema deriving a property for all ordinals from its validity whenever it holds at every earlier ordinal. This formal statement captures the central rule used in transfinite induction.
Natural number induction: A proof principle establishing a property for every natural number from a base case and a successor step. Ordinary induction is the finite-stage special case that transfinite induction extends.
Cantor normal form: A unique representation of each ordinal as a finite sum of decreasing powers of ω with positive integer coefficients. Existence and uniqueness proofs proceed by analyzing ordinal stages.
Transfinite induction on a set: Induction along a well-ordered set rather than across the class of all ordinals. It applies the same reasoning to a bounded well-order instead of every ordinal.
Successor ordinal: An ordinal of the form α + 1, immediately following the ordinal α. At successor stages, the earlier-stage assumption includes the immediate predecessor.
Ordinal arithmetic: Arithmetic operations on ordinals, whose results depend on the order of their terms. Ordinal arithmetic names stages and bounds that occur in transfinite proofs.
Borel hierarchy: A classification of sets generated from open sets through iterated complementation and countable union. Its levels are indexed by countable ordinals, making transfinite induction a natural proof tool.
Structural induction: A proof method establishing a property for recursively constructed objects from its validity on their components. It follows construction structure, while transfinite induction follows ordinal order.
Limit ordinal: A nonzero ordinal that is not a successor ordinal. Limit stages have no immediate predecessor, so the proof must use all earlier stages.
Set-theoretic universe: The totality of sets considered within a chosen foundation for set theory. The principle is commonly formulated for properties of ordinals in set theory.

Linked from 6 pages