Theorem

Transfinite Recursion Theorem

Logic and Foundations

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

Existence Theorem 1

Statement Form

Uniqueness Theorem 1

Statement Form

Inequality 1

Statement Form

Identity or Equation 1

Statement

The 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

In Branch

Set Theory

Source Transfinite Recursion Theorem (Wikipedia)

Sources

1. Transfinite Recursion Theorem (Wikipedia)

Wikimedia Foundation

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.
In Group: Set Theory, Lead section
sets. Since each well-ordered set is isomorphic to an ordinal, the theorem is also often stated in terms of
In Branch: Set Theory, Lead section
well-ordered set

Switch Atlas

The Inner World

The Civil World

The Natural World

Transfinite Recursion Theorem

Has Statement Form

In Branch