Fixed-Point Lemma for Normal Functions

Introduction

The Fixed-Point Lemma for Normal Functions is a fundamental result in set theory, particularly in the study of ordinal numbers and their functions. The lemma states that any normal function from the class of ordinals to itself has a proper class of fixed points which is unbounded in the ordinals.

Definitions and Preliminaries

Definition 1 (Normal Function)

A function f: Ord → Ord is normal if and only if it satisfies both:

1. (Strictly Increasing) For all ordinals α, β: if α < β, then f(α) < f(β)
2. (Continuous) For all limit ordinals λ: f(λ) = sup{f(α) | α < λ}

Definition 2 (Fixed Point)

Given a function f: Ord → Ord, an ordinal α is called a fixed point of f if f(α) = α.

Definition 3 (Class of Fixed Points)

For a normal function f, we define Fix(f) = {α ∈ Ord | f(α) = α}.

Main Theorem

Theorem 1 (Fixed-Point Lemma). Let f: Ord → Ord be a normal function. Then:

1. Fix(f) is non-empty
2. Fix(f) is unbounded in Ord (i.e., for any ordinal β, there exists α ∈ Fix(f) such that β < α)

Proof

We present two proofs of this result:

Proof 1 (Constructive Approach)

Lemma 1.1: For any ordinal β, we can construct a fixed point α > β.

Proof:

1. Define a sequence (αξ)ξ∈Ord recursively:

• α₀ = β
• αξ₊₁ = f(αξ) for successor ordinals
• αλ = sup{αξ | ξ < λ} for limit ordinals λ

1. By transfinite induction, prove that this sequence is strictly increasing:

• Base case: α₁ = f(α₀) > α₀ (by strictly increasing property)
• Successor step: αξ₊₁ = f(αξ) > αξ
• Limit step: For limit λ, αλ = sup{αξ | ξ < λ} > αξ for all ξ < λ

1. Let γ = sup{αξ | ξ ∈ Ord}. Then:
   f(γ) = f(sup{αξ}) (by definition)
   = sup{f(αξ)} (by continuity)
   = sup{αξ₊₁}
   = γ

Therefore, γ ∈ Fix(f) and γ > β.

Proof 2 (Via Iteration)

An alternative proof uses the iteration of f:

1. Define f⁰(α) = α
2. For any ordinal ξ, define:

• f^(ξ+1)(α) = f(f^ξ(α))
• f^λ(α) = sup{f^ξ(α) | ξ < λ} for limit λ

1. For any α, the sequence (f^ξ(α))ξ∈Ord eventually reaches a fixed point.

Applications

Theorem 2

The class Fix(f) of a normal function f is itself closed and unbounded in Ord.

Proof:

1. Closedness: Let (αᵢ)ᵢ∈I be an increasing sequence in Fix(f). Let λ = sup{αᵢ}. Then:
   f(λ) = sup{f(αᵢ)} = sup{αᵢ} = λ
2. Unboundedness: Shown in main theorem.

Common Errors and Misconceptions

1. Domain Confusion: The lemma specifically requires f: Ord → Ord. It does not generally apply to functions on proper sets or other domains.
2. Continuity vs. Monotonicity: Both properties are essential. A function that is merely strictly increasing or merely continuous may not have any fixed points.
3. Size of Fix(f): Fix(f) is always a proper class, not merely an unbounded set.

Notes

1. This lemma generalizes to certain larger domains in proper class theory.
2. The constructive proof gives an explicit method for finding fixed points.
3. The result is optimal in the sense that we cannot strengthen “unbounded” to “continuous” in general.

References

1. Jech, T. (2003). Set Theory: The Third Millennium Edition, revised and expanded. Springer.
2. Kanamori, A. (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer.

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