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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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