终极L（数学论文）一 (14-8)

(1) N＝L(N) ∩Vλ₊₂ and crit(j)＜λ；

(2) Nλ ⊆ L(N)；

(3)for all F：Vλ₊₁ → N \ {∅} such that F ∈ L(N) there exists G：Vλ₊₁ → Vλ₊₁ such that G ∈ N and such that for all A ∈ Vλ₊₁，G(A) ∈ F(A).

We shall state one claim that makes reference to Laver’s axiom at the end of Section 6，but shall not refer to it further duringthis section.

Definition 3.2.We define the sequence〈E⁰α(Vλ₊₁)：α＜ΥVλ₊₁〉to be the maximum sequence such that the following hold.

(1)E⁰₀(V₊₁)＝L(V₊₁)∩Vλ₊₂ and E⁰₁(Vλ₊₁)＝L((Vλ₊₁)#)∩Vλ₊₂.

(2)Suppose α＜ΥVλ₊₁ and α is a limit ordinal. Then E⁰α(Vλ₊₁)＝L(U {E⁰ᵦ(Vλ₊₁)：β＜α})∩Vλ₊₂.

(3)Suppose α＋1＜ΥVλ₊₁. Then for some X ∈ E⁰α₊₁(Vλ₊₁)，E⁰α(Vλ₊₁)＜X，where by this we mean that there is a surjection π：Vλ₊₁ → E⁰α(Vλ₊₁) with π ∈ L(X，Vλ₊₁)，and B⁰α₊₁(Vλ₊₁)＝L(X，Vλ₊₁)∩Vλ₊₂，and if α＋2＜ΥVλ₊₁ then E⁰α₊₂(Vλ₊₁)＝L((X，Vλ₊₁)#)∩Vλ₊₂.

(4)Suppose α＜Υλ₊₁. Then there exists X ⊆ Vλ₊₁ such that E⁰α(Vλ₊₁) ⊆ L(X，Vλ₊₁) and such that there is a proper elementary em-bedding j：L(X，Vλ₊₁) ≺ L(X，Vλ₊₁)，where this means that j is non-trivial with critical point below λ，and for all X' ∈ L (X，Vλ₊₁)∩Vλ₊₂ there exists a Y ∈ L(X，Vλ₊₁)∩Vλ₊₂ such that 〈Xᵢ：i＜ω〉∈L(Y，Vλ₊₁)，where X₀＝X' and Xᵢ₊₁＝j(Xᵢ) for all i ≥ 0.

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

(a) (cof(𝚹ᴺ))ᴸ⁽ᴺ⁾＜λ，or

(b)(cof(𝚹ᴺ))ᴸ⁽ᴺ⁾＞λ and for some Z ∈ N，L(N)＝(HODVλ₊₁∪{Z})ᴸ⁽ᴺ⁾.

Here 𝚹ᴺ＝sup{𝚹ᴸ⁽ˣ，ⱽλ⁺¹⁾：X ∈ N} where 𝚹ᴸ⁽ˣ，ⱽλ⁺¹⁾ is the supremum of the ordinals γ which can serve as the codomain of a suriection with domain Vλ₊₁ where the surjection is an element of L(X，Vλ₊₁).

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