Fixed-point lemma for normal functions

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The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117). It was first proved by Oswald Veblen in 1908.

Background and formal statement[edit]

A normal function is a class function f from the class Ord of ordinal numbers to itself such that:

  • f is strictly increasing: f(α) < f(β) whenever α < β.
  • f is continuous: for every limit ordinal λ (i.e. λ is neither zero nor a successor), f(λ) = sup { f(α) : α < λ }.

It can be shown that if f is normal then f commutes with suprema; for any nonempty set A of ordinals,

f(sup A) = sup {f(α) : α ∈ A }.

Indeed, if sup A is a successor ordinal then sup A is an element of A and the equality follows from the increasing property of f. If sup A is a limit ordinal then the equality follows from the continuous property of f.

A fixed point of a normal function is an ordinal β such that f(β) = β.

The fixed point lemma states that the class of fixed points of any normal function is nonempty and in fact is unbounded: given any ordinal α, there exists an ordinal β such that β ≥ α and f(β) = β.

The continuity of the normal function implies the class of fixed points is closed (the supremum of any subset of the class of fixed points is again a fixed point). Thus the fixed point lemma is equivalent to the statement that the fixed points of a normal function form a closed and unbounded class.

Proof[edit]

The first step of the proof is to verify that f(γ) ≥ γ for all ordinals γ and that f commutes with suprema. Given these results, inductively define an increasing sequence <αn> (n < ω) by setting α0 = α, and αn+1 = fn) for n ∈ ω. Let β = sup {αn : n ∈ ω}, so β ≥ α. Moreover, because f commutes with suprema,

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

The last equality follows from the fact that the sequence <αn> increases.

Example application[edit]

The function f : Ord → Ord, f(α) = ωα is normal (see initial ordinal). Thus, there exists an ordinal θ such that θ = ωθ. In fact, the lemma shows that there is a closed, unbounded class of such θ.

References[edit]