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

114 the realm of the infiniteThe definition of a supercompact cardinal is due to Reinhardt and Solovay (seeKanamori 1994) for more on the history of the axiom. Below is a reformulation of thedefinition attributed to Magidor in terms of extenders.Definition 26. A cardinal δ is a supercompact cardinal if for each ordinal β>δthere exists an extenderE : V α+1 → V β+1such that E(κ) = δ, where κ = CRT(E).Slightly stronger is the notion that δ is an extendible cardinal: for all α>δthereexists an extenderE : V α+1 → V β+1such that CRT(E) = δ.As I have already indicated, the strongest large cardinal axioms not known to beinconsistent with the Axiom of Choice are the family of axioms asserting the existenceof strongly (ω + 1)-huge cardinals. These axioms have seemed so far beyond anyconceivable inner model theory that they simply are not understood.The possibilities for an inner model theory at the level of supercompact cardinalsand beyond have been essentially a complete mystery until recently. The reason lies inthe nature of extenders. Again for expository purposes, let me define an extenderE : V α+1 → V β+1to be a suitable extender if E(CRT(E)) >α. Thus, an extenderE : V α+1 ∩ M → V β+1 ∩ Mis a suitable extender if it is not too long and if V β+1 ⊂ M. For example, supposethat κ is a strongly (ω + 1)-huge cardinal as defined in Definition 6. Then there existsγ>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+1Thus, j is an extender, but not a suitable extender. In particular, the existence of astrongly (ω + 1)-huge cardinal cannot be witnessed by a suitable extender.The Inner Model Program at the level of supercompact cardinals and beyond seeksenlargements N of L such that there are enough extenders E such that E|N ∈ N towitness that the targeted large cardinal axiom holds in N. For the weaker large cardinalaxioms, this has been an extremely successful program. For example, Mitchell andSteel (1994) have defined enlargements at the level of Woodin cardinals. In fact, theydefine the basic form of such enlargements of L up to the level of superstrong cardinals,which are just below the level of supercompact cardinals. The Mitchell-Steel modelsare constructed from sequences of extenders. The basic methodology is to construct N