Normal function

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title: "Normal function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["set-theory", "ordinal-numbers"] description: "Function of ordinals in mathematics" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Normal_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Function of ordinals in mathematics ::

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:

1. For every limit ordinal γ (i.e. γ is neither zero nor a successor), it is the case that {{math|1=f(γ) = sup{{mset|f(ν) : ν
2. For all ordinals {{math|α

Examples

A simple normal function is given by (see ordinal arithmetic). But is not normal because it is not continuous at any limit ordinal (for example, f(\omega) = \omega+1 \ne \omega = \sup {f(n) : n ). If β is a fixed ordinal, then the functions , (for β ≥ 1), and (for β ≥ 2) are all normal.

More important examples of normal functions are given by the aleph numbers f(\alpha) = \aleph_\alpha, which connect ordinal and cardinal numbers, and by the beth numbers 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 {{math|f(γ)

Furthermore, for any non-empty set S of ordinals, we have :. Proof: "≥" follows from the monotonicity of f and the definition of the supremum. For "≤", consider three cases:

• if , then and ;
• if is a successor, then is in S, so is in f(S), i.e. f(sup S) ≤ sup f(S);
• if is a nonzero limit, then for any {{math|ν

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. For a hierarchy of normal functions, see Veblen functions.

Notes

References

• {{citation |first=Peter |last=Johnstone |authorlink=Peter Johnstone (mathematician) |year=1987 |title=Notes on Logic and Set Theory |publisher=Cambridge University Press |isbn=978-0-521-33692-5 |url-access=registration |url=https://archive.org/details/notesonlogicsett0000john

References

1. {{harvnb. Johnstone. 1987
2. {{harvnb. Johnstone. 1987

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