终极L（数学论文）一 (14-2)

One can introduce the notion of a hyper-enormous* cardinal，of somewhat less strength than a hyper-enormous cardinal, and it can be shown that a cardinal κ which is a critical point of an clementary embedding j：Vλ＋₂ ≺ V＋₂，in a context not assuming choiceis necessarily a hyper-enormous* cardinal.(It is quitelikely that assuming the Axiom of Depending Choice it can be shown to be hyper-enormous，too，but the former proposition is all that is needed for what follows.) Building on this insight，we can obtain the result that the existence of such an elementary embedding is in fact outright inconsistent with ZF.

Finally,the assertion that there is a proper class of α-enormous cardinals for each limit ordinal α＞0 can be shown to imply a version of the Ultimate-L Conjecture.

Keywords：Ultimate-L program，large cardinals. MSC：03E45，03E55

To my beloved ωife Mαri Mnαtsαkαnyαn，ωithout ωhom this ωork

ωould not hαυe been possible.

NEW LARGE-CARDINAL AXIOMS AND THE ULTIMATE-L PROGRAM 3

ACKNOWLEDGEMENTS

Hugh Woodin provided very helpful feedback on a number of early drafts of this work in which a number of unsatisfactory definitions of the notion of an α-enormous cardinal were formulated，and I am very thankful for his assistance.

In what follows we will present a number of new large-cardinal ax- ioms,and applications of them.Let us begin by presenting the defini- tions of the newlarge-cardinal properties tobeconsidered.

FINITIONS OF THE NEW LARGE-CARDINAL PROPERTIES

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