终极L（数学论文）二 (7-1)

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10 MᶜCALLUM

from a set of assumptions which are provably consistent byforcing relative only to ZF plus the existence of an elementary embedding Vλ₊₂ ≺ Vλ₊₂. Thus the existence of an elementary embedding Vλ₊₂ ≺ Vλ₊₂ is in fact inconsistent with ZF. ▢

6.A PROOF OF THE ULTIMATE-L CONJECTURE

In this section，we will seek to give a proof of Hugh Woodin's Ultimate-L Conjecture.The most important sources for Hugh Woodin’s Ultimate-L program are [1]，[2]，and [3]. We must begin by giving the statement of the axiom V＝Ultimate-L，following Definition 7.14 of [3].

Definition 6.1. The axiom V＝Ultimate-L is defined to be the asser-tion that

(1)There is a proper class of Woodin cardinals.

(2) Given any Σ₂-sentence ф which is true in V，there exists a univer-sally Baire set of reals A，such that，if 𝚹ᴸ⁽ᴬ，ℝ⁾ is defined to be the least ordinal 𝚹 such that there is no surjection from ℝ onto 𝚹 in L(A，ℝ)，then the sentence ф is true in HODᴸ⁽ᴬ，ℝ⁾∩V𝚹ᴸ₍ᴀ，R₎.

Now let us recall a set of definitions from [3].

Definition 6.2. Suppose that N is a transitive proper class model of ZFC and that δ is a supercompact cardinal in V. We say that N is a weak extender model for δ supercompact，if for all γ＞δ，there exists on Pδ(γ) a normal fine δ-complete measure μ，with μ (N∩Pδ(λ))＝1and μ∩N ∈ N.

Definition 6.3. A sequence N：＝ 〈Nα：α ∈ Ord〉is weakly Σ₂-definable if there is a formula ф(x)such that

(1)For all β＜η₁＜η₂＜η₃， if (Nф)ⱽη₁│β＝(Nф)ⱽη₃│β then (Nф)ⱽη₁│β＝(Nф)ⱽη₂│β＝(Nф)ⱽη₃│β；

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