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: f (γ) = sup . 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 is the set , 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, which connect ordinal and cardinal numbers, and by the beth numbers.

Properties

If f is normal, then for any ordinal α, f (α) ≥ α . 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: δ = 0 , then S = and sup f (S) = f (0) 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) 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 ≤ 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. For a hierarchy of normal functions, see Veblen functions.