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

the infinite realm 115from a sequence of extenders that includes enough extenders to directly witness thatthe targeted large cardinal axiom holds in N.For the construction of the Mitchell-Steel models there is a fundamental requirementthat the extenders on sequence from which the enlargement of L is constructed bederived from extendersE : V α+1 ∩ M → V β+1 ∩ Msuch that V α+1 ⊂ M. Following this basic methodology, the enlargements of L atthe level of supercompact cardinals and beyond must be constructed from extendersequences, which now include extenders that are restrictions of extenders of the formE : V α+1 → V β+1 ,and Steel has shown that the basic methodology of analyzing extender models encountersserious obstructions once there are such extenders on the sequence, particularly ifthe extenders are not suitable.But by some fairly recent theorems something completely unexpected and remarkablehappens. Suppose that N is a transitive class, for some cardinal δ,N “δ is a supercompact cardinal,”and that this is witnessed by the class of all E|N such that E|N ∈ N and such that E is asuitable extender. Then the transitive class N is close to V, and N inherits essentially alllarge cardinals from V. The amazing thing is that this must happen no matter how N isconstructed. This would seem to undermine my earlier claim that inner models shouldbe constructed from extender sequences that contain enough extenders to witness thatthe targeted large cardinal axiom holds in the inner model. It does not, and the reasonis that by simply requiring that E|N ∈ N for enough suitable extenders from V towitness that the large cardinal axiom, “There is a supercompact cardinal,” holds in N,one (and this is the surprise) necessarily must have E|N ∈ N for a much larger class ofextenders, E : V α+1 → V β+1 . Therefore, the principle that there are enough extendersin N to witness that the targeted large cardinal axiom holds in N is preserved (as itmust be). The change, in the case that N is constructed from a sequence of extendersthat includes restrictions of suitable extenders, is that these extenders do not have to beon the sequence from which N is constructed. In particular in this case, large cardinalaxioms can be witnessed to hold in N by “phantom” extenders; these are extenders ofN that are not on the sequence and that cannot be witnessed to hold by any extenderon the sequence. This includes large cardinal axioms at the level of strongly (ω + 1)-huge cardinals. As a consequence of this, one can completely avoid the cited obstaclesbecause one does not need to have the kinds of extenders on the sequence that give riseto the obstacles. Specifically, one can restrict consideration to extender sequences ofjust extenders derived from suitable extenders, and this is a paradigm shift in the wholeconception of inner models.The analysis yields still more. Suppose that there is a positive solution (in ZFC) tothe inner model problem for just one supercompact cardinal. Then as a corollary onewould obtain a proof of the following conjecture:Conjecture (ZF).There are no weak Reinhardt cardinals.