终极L（数学论文）二 (7-6)

We should also note that if.V＝Ultimate-L，then if κ is ω-enormous as witnessed by〈κᵢ：i＜ω〉then Vκ₀ models the assertion that there is a proper class of λ satisfying Laver’s axiom，as defined in Section 3，and if κ is virtually ω-enormous as witnessed by〈κᵢ：i＜ω〉then Vκ₀ models the assertion that there is a proper class of Ramsey cardinals.

7. CONCLUDING REMARKS

The new large cardinals were inspired by Victoria Marshall’s work on reflection principles in[5] and are plausibly the correct generali-sation of the reflection principles which were demonstrated by her in that work to imply the existence of n-huge cardinals. The large cardi-nal axiom used to prove the Ultimate-L conjecture certainly has quite substantial consistency strength and some skepticism about its consis-tency would certainly be quite reasonable at this stage，but it may be

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM 13

that the further study of the inner model theory of Ultimate-L and inner models which approximate it from within will provide new in- sights and increased confidence in consistency. In the mean may very well be that the Ultimate-L conjecture is provable from just an extendible cardinal as originally envisaged by Hugh Woodin. so in that sense much work remains to be done.

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