Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

Theorem 5 (Kunen).beyond the finite realm 99There are no weak Reinhardt cardinals.The proof is elementary, so this does not refute the Skeptic’s Retreat, but Kunen’sproof makes essential use of the Axiom of Choice. The problem is open without thisassumption. Further, there is really no known interesting example of a strengtheningof the definition of a weak Reinhardt cardinal that yields a large cardinal axiom thatcan be refuted without using the Axiom of Choice. The difficulty is that without theAxiom of Choice it is extraordinarily difficult to prove anything about sets.Kunen’s proof leaves open the possibility that the following large cardinal axiommight be consistent with the Axiom of Choice. This, therefore, is essentially thestrongest large cardinal axiom not known to be refuted by the Axiom of Choice (seeKanamori [1994] for more on this, as well as for the actual statement of Kunen’stheorem).Definition 6. A cardinal κ is a strongly (ω + 1)-huge cardinal if there existγ>λ>κsuch that(1) V κ ≺ V λ ≺ V γ ,(2) there exists an elementary embeddingsuch that κ is the critical point of j.j : V λ+1 → V λ+1The issue of whether the existence of a weak Reinhardt cardinal is consistent withthe axioms ZF is an important issue for the Set Theorist because by the results ofWoodin (2009) the theoryZF + “There is a weak Reinhardt cardinal”proves the formal consistency of the theoryZFC + “There is a proper class of strongly (ω + 1)-huge cardinals.”This number theoretic statement is a theorem of number theory. As indicated previously,the notion of a strongly (ω + 1)-huge cardinal is essentially the strongest large cardinalnotion that is not known to be refuted by the Axiom of Choice.Therefore, the number theoretic assertion that the theoryZF + “There is a weak Reinhardt cardinal”is consistent is a stronger assertion than the number theoretic assertion that the theoryZFC + “There is a proper class of strongly (ω + 1)-huge cardinals”is consistent. More precisely, the former assertion implies, but is not implied by, thelatter assertion, unless, of course, the theoryZFC + “There is a proper class of strongly (ω + 1)-huge cardinals”is formally inconsistent. This raises an interesting question:How could the Set Theorist ever be able to argue for the prediction that the existence ofweak Reinhardt cardinals is consistent with axioms of set theory without the Axiom ofChoice?