Fixed-point lemma for normal functions

From Academic Kids

The fixed-point lemma for normal functions is a basic result in axiomatic set theory; it states that any normal function has arbitrarily large fixed points and can often be used to construct ordinal numbers with interesting properties. A formal version and proof (using the Zermelo-Fraenkel axioms) follow.

Formal version

Let f : Ord → Ord be a normal function. Then for every α ∈ Ord, there exists a β ∈ Ord such that β ≥ α and f(β) = β.

Proof

We know that f(γ) ≥ γ for all ordinals γ. We now declare an increasing sequence <αn> (n < ω) by setting α0 = α, and αn+1 = fn) for n < ω, and define β = sup <αn>. Clearly, β ≥ α. Since f commutes with suprema, we have

f(β) = f(sup {αn : n < ω})
       = sup {fn) : n < ω}
       = sup {αn+1 : n < ω}
       = β

(The last step uses the fact that the sequence <αn> increases).

Example application

It is easily checked that the function f : Ord → Ord, f(α) = אα is normal (see aleph number); thus, there exists an ordinal Θ such that Θ = אΘ. In fact, the above lemma shows that there are infinitely many such Θ.

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