Transfinite Recursion Theorem

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.