The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117).
It was first proved by Oswald Veblen in 1908.
A normal function is a class function
from the class Ord of ordinal numbers to itself such that: It can be shown that if
is normal then
commutes with suprema; for any nonempty set
is a successor ordinal then
and the equality follows from the increasing property of
is a limit ordinal then the equality follows from the continuous property of
A fixed point of a normal function is an ordinal
The fixed point lemma states that the class of fixed points of any normal function is nonempty and in fact is unbounded: given any ordinal
, there exists an ordinal
β ≥ α
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.
The first step of the proof is to verify that
commutes with suprema.
Given these results, inductively define an increasing sequence
β ≥ α
commutes with suprema, The last equality follows from the fact that the sequence
found in this way is the smallest fixed point greater than or equal to
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.