A New Fine-Structure Theory for Constructible Inner Models

Peter Koepke, Bonn London 25.04.2001

I. A Reminder of Gödel’s Model L II. The New Hierachy III. A ”Proof” of Jensen’s Covering Theorem I. A Reminder of Gödel’s Model L

L = _{α∈On}^{⋃} L_{α}, L_{0} = ∅

L_{α+1} L_{α}^{}

The Interpretation Function • I(L_{α}, φ, y⃗) = {x ∈ L_{α}|(L_{α}, ∈) ⊧ φ[x, y⃗]} • L_{α+1} = {I(L_{α}, φ, y⃗)|φ formula, y⃗ ∈ L_{α}} • λ limit → L_{λ} = _{α<λ}^{⋃} L_{α} The Gödel Hierachy satisfies

• α < β → L_{α} ⊆ L_{β}

• α < β → L_{α} ∈ L_{β}

• L_{α} transitive

• L ⊧ Zermelo-Fraenkel set theory Wellordering L

a <_{L} b :≡ ”least” triple (L_{α}, φ, y⃗) such that a = I(L_{α}, φ, y⃗) is lexicographically smaller than ”least” triple (L_{β}, φ, z⃗) such that b = I(L_{β}, ψ, z⃗).

• L ⊧ Axiom of Choice Skolem function

S(L_{α}, φ, y⃗) ≃ the <_{L} − least x such
that L_{α} ⊧ φ[x, y⃗]

• L ⊧ Generalized Continuum Hypothesis • L ⊧ ◇ Jensen’s Fine-Structure Theory

• I, <_{L}, S are definable in low complexity: S(L_{α}, φ, −) for Σ_{1}-fomulas φ is Σ_{1}(L_{α})

• Σ_{n}(L_{α}) is analysed as n iterations of Σ_{1}-definability.

Jensen • L ⊧ □

• L ⊧ Morasses exist

• Covering Theorem for L

II. The New Hierachy

The Approach

• Make I, <, S, ∈ symbols of a new language S.

• Define hierachy F_{α} = (F_{α}, I ↾ F_{α}, <↾ F_{α}, S ↾ F_{α}, ∈).

• F_{α+1} is obtained by quantifier-free S-definability.

The Hierachy S_{0} = quantifier-free S formulae I(F_{α}, φ, y⃗) = {x ∈ F_{α}|F_{α} ⊧ φ[x, y⃗]} F_{α+1} = {I(F_{α}, φ, y⃗)|φ ∈ S_{0}, y⃗ ∈ F_{α}} (Ternary) Wellordering and Skolem Function

a <_{F_{α}} b :≡ lex. least (F_{β}, φ, y⃗) with

a = I(F_{β}, φ, y⃗) is lex. smaller than lex. least (F_{γ}, ψ, z⃗) with b = I(F_{γ}, ψ, z⃗).

S(F_{α}, φ, y⃗) ≃ the <_{F_{α}}-least x such

that F_{α} ⊧ φ[x, y⃗] The Fine Hierachy

• F_{0} = (∅, …)

• F_{α+1} =(F_{α+1} , I ↾F_{α+1} , S ↾F_{α+1} , <↾F_{α+1} , ∈)

• limit λ → F_{λ} = _{α<λ}^{⋃} F_{α}

• L = _{α∈On}^{⋃} F_{α} Properties of the Fine Hierachy (F_{α})_{α∈On}

• hierachical properties as for L

• F_{ωα} = J_{α} (Jensen hierachy)

• Condensation:

If X ⊆ F_{α} is a substructure of F_{α} then X ≅ Fᾱ for some ᾱ ≤ α Direct Limits

If the directed System (F_{α_{i}}), (π_{ij}) of injective homomorphisms π_{ij} : F_{α_{i}} → F_{α_{j}} has a wellfounded limit, the limit is isomorphic to some F_{α}.

III. A ”Proof” of Jensen’s Covering Theorem

The Covering Theorem

Assume O^{#} does not exist. Then L covers V, i.e.,

∀X ⊆ On ∃Z ∈ L (X ⊆ Z ∧ ¯ Z̄ ≤ ¯ X̄ + ℵ_{1})

Corollary

If O^{#} does not exist, then

• every singular cardinal κ in V is singular in L

• moreover: (κ^{+} )^{V} = (κ^{+} )^{L} ”Proof” (by Contradiction)

Let X be minimal counterexample to covering: τ = sup (X) minimal, card (X) minimal.

O^{#} ≃ there exists a non-trivial elementary L → L.

Structure of the Proof The Extension of F_{δ} by σ

[figure: L; L; σ̃; ⊆; ⊆; Fτ̄; F_{τ}; σ; Y; X; ≅; ≺; ⊆]

[figure: F_{δ}; Fδ̃; σ̃; ⊆; ⊆; Fτ̄; F_{τ}; σ]

ASSUMPTION

∀μ < τ̄ ∀ finite p ⊆ F_{δ}: the substructur of F_{δ} generated by μ ∪ p is isomorphic to some F_{α(μ,p)}, α(μ, p) ≤ τ̄

π_{μp} : F_{α(μ,p)} ≅ ⟨μ ∪ p⟩ ⊆ F_{δ}

Lifting a Directed System

[figure: F_{δ}; Fδ̃; σ̃ ⊇ σ; F_{α(ν,q)}; F_{α^{∗}(ν,q)}; F_{α(μ,p)}; F_{α^{∗}(μ,p)}; ↑ π_{μp,νq}; ↑ π_{μp,νq}^{∗}; ↑ π_{νp}; ↑ π_{νq}^{∗}; ⟼; σ; ∈ Fτ̄; ∈ F_{τ}]

If δ = ∞ satisfies the ASSUMPTION then 0^{#} exists, CONTRADICTION.

Otherwise let δ be maximal satisfying the ASSUMPTION.

• F_{δ+1} is generated from some ν ∪ q

F_{δ+1} = ⟨ν ∪ q⟩

Pushing up a ∃_{1}-Elementary Map

[figure: σ̃ : F_{δ}; Fδ̃; ⊆; ⊆; σ̃^{+} : F_{δ+1}; F_{δ̃+1}; is ∃_{1}-elementary; is a homorphism]

X ⊆ range (σ) ⊆ range (σ̃^{+} )
= σ̃^{+} [⟨ν ∪ q⟩]
= ⟨σ̃^{+} [ν] ∪ σ̃^{+} [q]⟩
⊆ ⟨σ̃^{+} (ν) ∪ σ̃^{+} [q]⟩ =: W

where W ∈ L, card ^{L}(W) ≤ σ̃^{+} (ν) < τ Get covering set for X, CONTRADICTION.

Further Applications

• □, Morasses (Hyperfine structure theory, Friedman, K.)

• Coremodels: σ̃ : F_{δ} → Fδ̃ corresponds to a fine-structural ultrapower at F_{δ} by the ”extender” σ.