The transfinite recursion theorem states that a function can be defined by recursion over a well-ordered set, extending the everyday recursion used to define a function step by step over the natural numbers so it also works over any well-ordered set, not only over the ordinary whole numbers. Because every well-ordered set is order-isomorphic to some ordinal number, the theorem's usually stated directly in terms of ordinals too, and it underlies constructions throughout set theory that build an object stage by stage across a possibly infinite, well-ordered sequence of stages.
Facts
Classification
Statement Form Statement Form Statement Form Statement Form StatementThe transfinite recursion theorem states that for a class function G there exists a unique transfinite sequence F defined on the ordinals such that the value of F at any ordinal alpha equals G applied to the restriction of F to every ordinal less than alpha, which lets a function be defined by recursion over any well ordered set and not only over the natural numbers. 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Transfinite Recursion Theorem (Wikipedia)
Wikimedia FoundationLedeQuote, Lede
In mathematics, the transfinite recursion theorem says a function can be defined using a recursion over a well-ordered set; for example, N but also over general well-ordered sets.
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