# Normal function

This article relies largely or entirely on a single source. (March 2024) |

In axiomatic set theory, a function *f* : Ord → Ord is called **normal** (or a **normal function**) if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:

- For every limit ordinal γ (i.e. γ is neither zero nor a successor), it is the case that
*f*(*γ*) = sup{*f*(*ν*) :*ν*<*γ*}. - For all ordinals
*α*<*β*, it is the case that*f*(*α*) <*f*(*β*).

## Examples

[edit]A simple normal function is given by *f* (*α*) = 1 + *α* (see ordinal arithmetic). But *f* (*α*) = *α* + 1 is *not* normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set {*λ* + 1} is the set {*λ*}, which is not open when λ is a limit ordinal. If β is a fixed ordinal, then the functions *f* (*α*) = *β* + *α*, *f* (*α*) = *β* × *α* (for *β* ≥ 1), and *f* (*α*) = *β*^{α} (for *β* ≥ 2) are all normal.

More important examples of normal functions are given by the aleph numbers , which connect ordinal and cardinal numbers, and by the beth numbers .

## Properties

[edit]If f is normal, then for any ordinal α,

*f*(*α*) ≥*α*.^{[1]}

**Proof**: If not, choose γ minimal such that *f* (*γ*) < *γ*. Since f is strictly monotonically increasing, *f* (*f* (*γ*)) < *f* (*γ*), contradicting minimality of γ.

Furthermore, for any non-empty set S of ordinals, we have

*f*(sup*S*) = sup*f*(*S*).

**Proof**: "≥" follows from the monotonicity of f and the definition of the supremum. For "≤", set *δ* = sup *S* and consider three cases:

- if
*δ*= 0, then*S*= {0} and sup*f*(*S*) =*f*(0); - if
*δ*=*ν*+ 1 is a successor, then there exists s in S with*ν*<*s*, so that*δ*≤*s*. Therefore,*f*(*δ*) ≤*f*(*s*), which implies*f*(δ) ≤ sup*f*(*S*); - if δ is a nonzero limit, pick any
*ν*<*δ*, and an s in S such that*ν*<*s*(possible since*δ*= sup*S*). Therefore,*f*(*ν*) <*f*(*s*) so that*f*(*ν*) < sup*f*(*S*), yielding*f*(*δ*) = sup {*f*(ν) :*ν*<*δ*} ≤ sup*f*(*S*), as desired.

Every normal function f has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof. One can create a normal function *f ′* : Ord → Ord, called the **derivative** of f, such that *f ′*(*α*) is the α-th fixed point of f.^{[2]} For a hierarchy of normal functions, see Veblen functions.

## Notes

[edit]**^**Johnstone 1987, Exercise 6.9, p. 77**^**Johnstone 1987, Exercise 6.9, p. 77