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.

A normal function is a class function ${\displaystyle f}$ from the class Ord of ordinal numbers to itself such that:

• ${\displaystyle f}$ is strictly increasing: ${\displaystyle f(\alpha )<f(\beta )}$ whenever ${\displaystyle \alpha <\beta }$.
• ${\displaystyle f}$ is continuous: for every limit ordinal ${\displaystyle \lambda }$ (i.e. ${\displaystyle \lambda }$ is neither zero nor a successor), ${\displaystyle f(\lambda )=\sup\{f(\alpha ):\alpha <\lambda \}}$.

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

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

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

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

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

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.

The first step of the proof is to verify that ${\displaystyle f(\gamma )\geq \gamma }$ for all ordinals ${\displaystyle \gamma }$ and that ${\displaystyle f}$ commutes with suprema. Given these results, inductively define an increasing sequence ${\displaystyle \langle \alpha _{n}\rangle _{n<\omega }}$ by setting ${\displaystyle \alpha _{0}=\alpha }$, and ${\displaystyle \alpha _{n+1}=f(\alpha _{n})}$ for ${\displaystyle n\in \omega }$. Let ${\displaystyle \beta =\sup _{n<\omega }\alpha _{n}}$, so ${\displaystyle \beta \geq \alpha }$. Moreover, because ${\displaystyle f}$ commutes with suprema,

${\displaystyle f(\beta )=f(\sup _{n<\omega }\alpha _{n})}$

${\displaystyle \qquad =\sup _{n<\omega }f(\alpha _{n})}$

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

${\displaystyle \qquad =\beta }$

The last equality follows from the fact that the sequence ${\displaystyle \langle \alpha _{n}\rangle _{n}}$ increases. ${\displaystyle \square }$

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

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.