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) whenever $\alpha<\beta$.
* $f$ is continuous: for every limit ordinal $\lambda$ (i.e. $\lambda$ is neither zero nor a successor), $f(\lambda)=\sup\$.
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\$.
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 $\langle\alpha_n\rangle_$ by setting $\alpha_0 = \alpha$, and $\alpha_ = f(\alpha_n)$ for $n\in\omega$. Let $\beta = \sup_ \alpha_n$, so $\beta\ge\alpha$. Moreover, because $f$ commutes with suprema,
:$f(\beta) = f(\sup_ \alpha_n)$
:$\qquad = \sup_ f(\alpha_n)$
:$\qquad = \sup_ \alpha_$
:$\qquad = \beta$
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) = 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

*
*{{cite journal
, author= Veblen, O.
, authorlink = Oswald Veblen
, title = Continuous increasing functions of finite and transfinite ordinals
, journal = Trans. Amer. Math. Soc.
, volume = 9
, year = 1908
, pages = 280–292
, id = Available vi
JSTOR

, doi= 10.2307/1988605
, issue = 3
, jstor= 1988605
, issn= 0002-9947, doi-access = free

Ordinal numbers
Normal Functions
Lemmas in set theory
Articles containing proofs