The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed pointss (Levy 1979: p.117). It was first proved by Oswald Veblen in 1908.

Background and formal statement

A normal function is a class from the class Ord of ordinal numbers to itself such that:

• is strictly increasing.

• is continuous.

commutes with suprema of ordinals,

1. .

.

A fixed point.

.

The continuity of the normal function implies the class of fixed points is closed (the supremum of any subset of the class of fixed points is again a fixed point). Thus the fixed-point lemma is equivalent to the statement that the fixed points of a normal function form a closed and unbounded class.

Proof

commutes with suprema,

.

Example application

The function f : Ord → Ord, f(α) = ωα is normal (see initial ordinal). Thus, there exists an ordinal θ such that θ = ωθ. In fact, the lemma shows that there is a closed, unbounded class of such θ.

References