New Large Cardinal Axioms and the Ultimate-L Program

We will consider a number of new large-cardinal properties, the $α$-tremendous cardinals for each limit ordinal $α>0$, the hyper-tremendous cardinals, the $α$-enormous cardinals for each limit ordinal $α>0$, and the hyper-enormous cardinals. For limit ordinals $α>0$, the $α$-tremendous cardinals and hyper-tremendous cardinals have consistency strength between I3 and I2. An $ω$-enormous cardinal has consistency strength greater than I0, and also all the large-cardinal axioms discussed in the second part of Hugh Woodin's paper on suitable extender models, not known to be inconsistent with ZFC and of greater consistency strength than I0. Ralf Schindler and Victoria Gitman have developed the notion of a virtual large-cardinal property, and a clear sense can be given to the notion of "virtually $ω$-enormous". A virtually $ω$-enormous cardinal can be shown to dominate a Ramsey cardinal. It can be shown that a cardinal $κ$ which is a critical point of an elementary embedding $j:V_{λ+2} \prec V_{λ+2}$, in a context not assuming choice, is necessarily a hyper-enormous cardinal. Building on this insight, we can obtain the result that the existence of such an elementary embedding is in fact outright inconsistent with ZF. The assertion that there is a proper class of $α$-enormous cardinals for every limit ordinal $α>0$ can be shown to imply a version of the Ultimate-L Conjecture.

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