Normal function

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

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 (for example,

f

(

ω

)

=

ω

+

1

≠

ω

=

sup

{

f

(

n

)

:

n

<

ω

}

{\displaystyle f(\omega )=\omega +1\neq \omega =\sup\{f(n):n<\omega \}}

). If β is a fixed ordinal, then the functions f (α) = β + α, f (α) = β × α (for β ≥ 1), and f (α) = βα (for β ≥ 2) are all normal.

More important examples of normal functions are given by the aleph numbers

f

(

α

)

=

ℵ

α

{\displaystyle f(\alpha )=\aleph _{\alpha }}

, which connect ordinal and cardinal numbers, and by the beth numbers

f

(

α

)

=

ℶ

α

{\displaystyle f(\alpha )=\beth _{\alpha }}

.

Properties

If f is normal, then for any ordinal α,

f (α) ≥ α.

Proof: If not, choose γ minimal such that f (γ) < γ.