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; 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) = 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.1Proof: 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
- Johnstone, Peter (1987), Notes on Logic and Set Theory, Cambridge University Press, ISBN 978-0-521-33692-5
References
Johnstone 1987, Exercise 6.9, p. 77 - Johnstone, Peter (1987), Notes on Logic and Set Theory, Cambridge University Press, ISBN 978-0-521-33692-5 https://archive.org/details/notesonlogicsett0000john ↩
Johnstone 1987, Exercise 6.9, p. 77 - Johnstone, Peter (1987), Notes on Logic and Set Theory, Cambridge University Press, ISBN 978-0-521-33692-5 https://archive.org/details/notesonlogicsett0000john ↩