Fixed-point lemma for normal functions
- Fixed-point lemma for normal functions
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
A normal function is a class function "f" from the class Ord of ordinal numbers to itself so that:
* "f" is increasing: "f"(α) ≤ f(β) whenever α ≤ β.
* "f" is continuous: for every limit ordinal λ, "f"(λ) = sup { f(α) : α < λ }.It can be shown that if "f" is normal then "f" commutes with suprema; for any set "A" of ordinals,:"f"(sup "A") = sup {"f"(α) : α ∈ "A" }.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
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 <α"n"> ("n" < ω) by setting α0 = α, and α"n"+1 = "f"(α"n") for "n" ∈ ω. Let β = sup {α"n" : "n" ∈ ω}, so β ≥ α. Moreover, because "f" commutes with suprema, :"f"(β) = "f"(sup {α"n" : "n" < ω}) : = sup {"f"(α"n") : "n" < ω} : = sup {α"n"+1 : "n" < ω} : = β The last equality follows from the fact that the sequence <α"n"> increases.
Example application
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 lemma shows that there is a closed, unbounded class of such Θ.
References
* cite book
author = Levy, A.
title = Basic Set Theory
year = 1979
publisher = Springer
id= Republished, Dover, 2002. ISBN 0-486-42079-5
*cite journal
author= Veblen, O.
authorlink = Oswald Veblen
title = Continuous increasing functions of finite and transfinite ordinals
journal = Trans. Amer. Math Soc.
volume = 9
year = 1908
pages = 280–292
id = Available via [http://links.jstor.org/sici?sici=0002-9947%28190807%299%3A3%3C280%3ACIFOFA%3E2.0.CO%3B2-1 JSTOR] .
doi= 10.2307/1988605
issue = 3
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