Proof of the Fixed-Point Theorem for Normal Functions

Every continuous strictly increasing function on ordinals has a proper class of fixed points. This result underlies the construction of epsilon numbers, aleph fixed points, and the Veblen hierarchy.

Statement

Theorem7.1Fixed-point theorem for normal functions

Let $f: \mathbf{Ord} \to \mathbf{Ord}$ be a normal function (strictly increasing and continuous). Then:

$f$ has a fixed point (i.e., $\exists \alpha: f(\alpha) = \alpha$ ).
The class of fixed points of $f$ is a proper class.
The enumeration of fixed points is itself a normal function.

Proof

Part 1: Existence of the least fixed point.

Define the sequence $\alpha_0 = 0$ , $\alpha_{n+1} = f(\alpha_n)$ for $n < \omega$ . Let $\alpha^* = \sup_{n < \omega} \alpha_n$ .

Since $f$ is continuous: $f(\alpha^*) = f(\sup_n \alpha_n) = \sup_n f(\alpha_n) = \sup_n \alpha_{n+1} = \sup_n \alpha_n = \alpha^*$ .

So $\alpha^*$ is a fixed point. (Note: if $f(0) = 0$ , then $\alpha^* = 0$ ; otherwise $\alpha^* > 0$ .)

Part 2: Proper class of fixed points.

Given any ordinal $\beta$ , we find a fixed point $> \beta$ . Start with $\gamma_0 = \beta + 1$ , $\gamma_{n+1} = f(\gamma_n)$ , $\gamma^* = \sup_n \gamma_n$ . By the same argument, $f(\gamma^*) = \gamma^*$ , and $\gamma^* \geq \gamma_0 > \beta$ .

Since for every $\beta$ there is a fixed point above $\beta$ , the class of fixed points is unbounded in $\mathbf{Ord}$ , hence a proper class.

Part 3: The enumeration is normal.

Let $g: \mathbf{Ord} \to \mathbf{Ord}$ enumerate the fixed points of $f$ in order: $g(\alpha)$ is the $\alpha$ -th fixed point. We verify:

Strictly increasing: Immediate from the definition (the fixed points are listed in order).
Continuous: For a limit $\lambda$ , $g(\lambda) = \sup_{\alpha < \lambda} g(\alpha)$ . Let $\gamma = \sup_\alpha g(\alpha)$ . Then $\gamma$ is a limit of fixed points. Since $f$ is continuous: $f(\gamma) = \sup_\alpha f(g(\alpha)) = \sup_\alpha g(\alpha) = \gamma$ . So $\gamma$ is a fixed point, and it is the supremum of all earlier fixed points, hence $g(\lambda) = \gamma$ . $\blacksquare$

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Applications

ExampleIterated fixed-point constructions

Starting with $f_0(\alpha) = \omega^\alpha$ (ordinal exponentiation with base $\omega$ ):

Fixed points of $f_0$ : the epsilon numbers $\varepsilon_0, \varepsilon_1, \ldots$ Enumerated by $f_1(\alpha) = \varepsilon_\alpha$ .
Fixed points of $f_1$ : ordinals $\alpha$ with $\varepsilon_\alpha = \alpha$ . These are the $\zeta$ -numbers. Enumerated by $f_2$ .
Continuing, one obtains the Veblen hierarchy $\varphi_\beta$ for all $\beta < \Gamma_0$ .

The process of taking fixed-point enumerations can itself be iterated transfinitely, leading to ever-larger ordinals.

RemarkConnection to large cardinals

The fixed-point theorem for normal functions applies to cardinal operations:

The function $\alpha \mapsto \aleph_\alpha$ is normal, with fixed points $\kappa = \aleph_\kappa$ .
The function $\alpha \mapsto \beth_\alpha$ is normal, with fixed points $\kappa = \beth_\kappa$ (strong limit cardinals).
Mahlo cardinals are inaccessible cardinals $\kappa$ such that the set of inaccessible cardinals below $\kappa$ is stationary -- a reflection of the fixed-point idea at the level of large cardinals.