终极L（数学论文）一 (14-9)

(6)Suppose α＋1＜TVλ₊₁，α is a limit ordinal，and let N＝E⁰α(Vλ₊₁).Then either

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM7

(a) (cof(𝚹ᴺ))ᴸ⁽ᴺ⁾＜λ，and E⁰α₊₁(Vλ₊₁)＝L(Nλ，N)∩Vλ₊₂，or

(b) (cof(𝚹ᴺ))ᴸ⁽ᴺ⁾＞λ，and E⁰α₊₁(Vλ₊₁)＝L(ε(N)，N)∩Vλ₊₂，where ε(N)is the set of elementary embeddings k：N ≺ N.

Define N：＝L(∪{E⁰α(Vλ₊₁)│α＜ΥVλ₊₁}) ∩ Vλ₊₂. Suppose that

cof(𝚹ᴺ)＞λ and L(N) ≠ (HODVλ₊₁∪{Z})ᴸ⁽ᴺ⁾ for all Z ∈ N，and further there is an elementary embedding j：L(N) ≺ L(N) with

crit(j)＜λ. Then we say that λ satisfies Woodin’s axiom.

Theorem 3.3.Suppose thαt κ is ω-enormous αs ωitnessed by the se-guence〈κₙ：n＜ω〉.Then Vκ₀ is α model for the αssertion thαt there is α proper clαss of λ sαtisfying Woodin’s αxiom.

Proof.Assume the hypotheses and notation given in the statement of the theorem. If we let λ：＝sup{κₙ：n＜ω}，then there is an elementary embedding j：Vλ₊₁ ≺ Vλ₊₁ with critical sequence〈κₙ：n＜ω〉.It is clearly sufficient to prove that λ satisfies Woodin’s axiom and can also beshown tosatisty Laver’s axiom on the assumption that V＝HOD，via embeddings extending j，in Vκ.

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