终极L（数学论文）二 (7-4)

3. Thus this model will still remain a model for the assertion that there is a proper class of α-enormous cardnals for each limit ordinal α＞0. Further clearly this inner model will satisfy GCH，and it is easily seen to be weakly Σ₂-definable and a subclass of HOD， and a weak extender model for the supercompactness of any cardinal δ which is supercompact in V，given that the stated large-cardinal hypothesis holds in V. In order to see the last point，it is necessary to observe that given the stated large-cardinal assumptions，any supercompact cardinal is necessarily hyper-enormous，and all necessary elementary

12 MᶜALLUM

embeddings for witnessing this do descend to the model Ultimate-L. We must now show that this model is indeed a model for the axiom V＝Ultimate-L as stated at the start of this section.

Clearly，our version of Ultimate-L is a model for the assertion that there is a proper class of Woodin cardinals. Suppose then，that some

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