终极L（数学论文）一 (14-10)

Suppose that a sequence of sets〈E⁰α(Vλ₊₁)：α＜β〉 satisfies require- ments (1)-(6) of the definition of Woodin’s axiom, relativized to Vκ，for some β ≤ ΥVλ₊₁，and define N to be the unique possible candi-date for E⁰ᵦ if it exists. It can be shown by transinite induction that L(j(N)∪Vλ₊₁)∩Vκ＝L(N)∩Vκ. nsidering that the action of j on the elements of such an N is determined by j│Vλ，and using the hypothesis of ω-enormousness，it can be shown by transfinite induc-tion that the restriction of j to L(N)∩Vκ is an elementary embedding L(N)∩Vκ ≺ L(N)∩Vκ，proper in the case where β＜ΥVλ₊₁. Since this is so for every N satisfying all the previously stated requirements we may now conclude by transfinite induction that λ satisfies Woodin’s axiom in Vκ via an embedding extending the restriction of j to Vλ₊₁.This completes the argument. ▢

This completes the demonstration that α-enormous and hyper-enormous cardinals have greater consistency strength than any previously con-sidered extension of ZFC not known to be inconsistent.

4. VIRTUALLY α-ENORMOUS AND HYPER-ENORMOUS CARDINALS

Ralf Schindler and Victoria Gitman in [4] have introduced the notion of virtual large-cardinal properties.Given any large-cardinal property defined with reference to a set-sized elementary embedding j：Vα ≺ Vᵦ or family of such embeddings，the corresponding virtual large-cardinal

8 MᶜALLUM

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