终极L（数学论文）二 (7-2)

(2)For all β ∈ Ord，N│β＝(Nф)ⱽη│β for sufficiently large η，where，for all γ， (Nф)ⱽγ＝{α ∈ Vᵧ：Vᵧ╞ф[α]}. Suppose N ⊂ V is an inner model such that N╞ ZFC. Then N is weakly Σ₂-definable if the sequence〈N∩Vα：α ∈ Ord〉is weakly Σ₂-deinable.

We can now state the result we plan to prove in this section.

Theorem 6.4.Suppose thαt there is α proper clαss of α-enormous cαrdinαls for eαch limit ordinαl α＞0.Then the folloωing υersion of the Ultimαte-L conjecture，giυen αs Conjecture 7.41 in [3]，holds.

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM 11

Suppose thαt δ is αn extendible cαrdinαl (in fαct one cαn eυen suppose only thαt δ is α supercompαct cαrdinαl).Then there is α ωeαk extender model N for the supercompαctness of δ such thαt

(1) N is ωeαkly Σ₂-definαble αnd N ⊂ HOD；

(2) N＝“V＝Ultimαte-L”.

(3) N ╞ GCH.

Proof of Theorem 6.4.Let us give the long awaited definition of Ultimate-L. We claim that what follows is the correct definition of Ultimate-L，assuming that there are sufficiently many large cardinals in V as out-lined in the hypotheses for Theorem 6.4.The correct way to deline it when we are making weaker large-cardinal assumptions still remains to be discovered.

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