Normal function

In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) iff 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), f(γ) = sup {f(ν) : ν < γ}.
  2. For all ordinals α < β, 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. 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(\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.[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:

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 g : Ord → Ord, called the derivative of f, where g(α) is the α-th fixed point of f.[2]

Notes

  1. Johnstone 1987, Exercise 6.9, p. 77
  2. Johnstone 1987, Exercise 6.9, p. 77

References

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