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:

• $f$ 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), $f(\lambda )=\sup \{f(\alpha ):\alpha <\lambda \}$.

It can be shown that if $f$ is normal then $f$ commutes with suprema; for any nonempty set $A$ of ordinals,

$f(\sup A)=\sup f(A)=\sup \{f(\alpha ):\alpha \in A\}$.

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 $\beta \ge \alpha$ 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 $⟨{\alpha }_n{⟩}_{n<\omega }$ by setting ${\alpha }_0=\alpha$, and ${\alpha }_{n+1}=f({\alpha }_n)$ for $n\in \omega$. Let $\beta =\underset{n<\omega }{\sup }{\alpha }_n$, so $\beta \ge \alpha$. Moreover, because $f$ commutes with suprema,

$f(\beta )=f(\underset{n<\omega }{\sup }{\alpha }_n)$

$=\underset{n<\omega }{\sup }f({\alpha }_n)$

$=\underset{n<\omega }{\sup }{\alpha }_{n+1}$

$=\beta$

The last equality follows from the fact that the sequence $⟨{\alpha }_n{⟩}_n$ increases. $◻$

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) = ω_α is normal (see initial ordinal). Thus, there exists an ordinal θ such that θ = ω_θ. In fact, the lemma shows that there is a closed, unbounded class of such θ.

References

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