Fixed-point lemma for normal functions

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.

Background and formal statement
A normal function is a class function $$f$$ from the class Ord of ordinal numbers to itself such that: It can be shown that if $$f$$ is normal then $$f$$ commutes with suprema; for any nonempty set $$A$$ of ordinals,
 * $$f$$ is strictly increasing: $$f(\alpha)<f(\beta)$$ whenever $$\alpha<\beta$$.
 * $$f$$ is continuous: for every limit ordinal $$\lambda$$ (i.e. $$\lambda$$ is neither zero nor a successor), $$f(\lambda)=\sup\{f(\alpha):\alpha<\lambda\}$$.
 * $$f(\sup A)=\sup f(A) = \sup\{f(\alpha):\alpha \in A\}$$.

Indeed, if $$\sup A$$ is a successor ordinal then $$\sup A$$ is an element of $$A$$ and the equality follows from the increasing property of $$f$$. If $$\sup A$$ is a limit ordinal then the equality follows from the continuous property of $$f$$.

A fixed point of a normal function is an ordinal $$\beta$$ such that $$f(\beta)=\beta$$.

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 $$\alpha$$, there exists an ordinal $$\beta$$ such that $$\beta\geq\alpha$$ and $$f(\beta)=\beta$$.

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
The first step of the proof is to verify that $$f(\gamma)\ge\gamma$$ for all ordinals $$\gamma$$ and that $$f$$ commutes with suprema. Given these results, inductively define an increasing sequence $$\langle\alpha_n\rangle_{n<\omega}$$ by setting $$\alpha_0 = \alpha$$, and $$\alpha_{n+1} = f(\alpha_n)$$ for $$n\in\omega$$. Let $$\beta = \sup_{n<\omega} \alpha_n$$, so $$\beta\ge\alpha$$. Moreover, because $$f$$ commutes with suprema,
 * $$f(\beta) = f(\sup_{n<\omega} \alpha_n)$$
 * $$\qquad = \sup_{n<\omega} f(\alpha_n)$$
 * $$\qquad = \sup_{n<\omega} \alpha_{n+1}$$
 * $$\qquad = \beta$$

The last equality follows from the fact that the sequence $$\langle\alpha_n\rangle_n$$ increases. $$ \square $$

As an aside, it can be demonstrated that the $$\beta$$ found in this way is the smallest fixed point greater than or equal to $$\alpha$$.

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.