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

98 the realm of the infiniteThis must be a cardinal. The critical point of j is the large cardinal, and the existenceof the transitive class M and the elementary embedding j are the witnesses for this.A cardinal κ is a measurable cardinal if there exists a transitive class M and anelementary embeddingj : V → Msuch that κ is the critical point of j.It is by requiring M to be closer to V that one can define large cardinal axiomsfar beyond the axiom, “There is a measurable cardinal.” In general, the closer onerequires M to be to V, the stronger the large cardinal axiom. The natural maximumaxiom was proposed (M = V ) by Reinhardt in his PhD thesis (see Reinhardt 1970).The associated large cardinal axiom is that of a Reinhardt cardinal.Definition 3.embeddingA cardinal κ is a Reinhardt cardinal if there is an elementarysuch that κ is the critical point of j.j : V → VThe definition of a Reinhardt cardinal makes essential use of classes, but the followingvariation does not, and this variation (which is not a standard notion) is onlyformulated in order to facilitate this discussion. The definition requires a logical notion.Suppose that α and β are ordinals such that αλ>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+2 → V λ+2The definition of a weak Reinhardt cardinal only involves sets. The relationshipbetween Reinhardt cardinals and weak Reinhardt cardinals is unclear, but one wouldnaturally conjecture that, at least in terms of consistency strength, Reinhardt cardinalsare stronger than weak Reinhardt cardinals, and hence my choice in terminology. Thefollowing theorem is an immediate corollary of the fundamental inconsistency resultsof Kunen (1971).