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

conclusions 117arguably a serious foundational issue for set theory for two reasons. First, these axiomsare known to refute the Axiom of Choice, and second, these axioms are known to be“stronger” than essentially all the notions of infinity believed to be formally consistentwith the Axiom of Choice. Here the metric for strength is simply the inference relationfor the corresponding predictions (of formal consistency). The issues raised by thisare twofold. First (regarding the debate between the Set Theorist and the Skeptic),there is no need to explain the success of a single prediction; it is a succession ofever-stronger successful predictions that demands explanation. But this one predictionof consistency subsumes all the predictions made to date, so there is no series ofpredictions that requires explanation. Second, for the Set Theorist to account for thisone prediction, it would seem that a different conception of the universe of sets isrequired.The conception of a universe of sets in which the Axiom of Choice fails creates moredifficulties than it solves, so this does not seem to be a viable option. However, anylarge cardinal axiom (which is expressible by a 2 -sentence) that can hold in a universeof sets satisfying all of the axioms except for the Axiom of Choice can hold in a genericextension of a universe of sets that does satisfy the Axiom of Choice. Therefore, thischallenge, as well as the challenge posed by formerly unsolvable problems such asthat of the continuum hypothesis, might be addressed (but perhaps not completely ina satisfactory manner) by adopting the conception of a multiverse of sets. Here the Conjecture emerges as a key conjecture. If this conjecture is true, then what isarguably the only candidate for a multiverse view for the infinite realm that can addressthese challenges also fails to be a viable alternative (accepting the requirement that themultiverse laws of Section 4.4 be satisfied). Therefore, if the multiverse view is correct,the Conjecture must be false.The attempt to understand how the Conjecture might be refuted leads directly tothe Inner Model Program. The Inner Model Program is the attempt to generalize thedefinition of L to yield transitive classes M in which large cardinal axioms hold. If theInner Model Program as described in the fifth section can be extended to the level ofa single supercompact cardinal, then no known large cardinal axiom can refute the Conjecture. Further, one would also obtain as corollary the verification of a series ofconjectures. These conjectures imply that the large cardinal axioms, such as the axiomthat asserts the existence of a weak Reinhardt cardinal, which pose such a challenge tothe conception of the universe of sets, are formally inconsistent. These inconsistencyresults would be the first examples of inconsistency results for large cardinal axiomsobtained only through a very detailed analysis.Finally, the extension of the Inner Model Program to the level of one supercompactcardinal will yield examples (where none are currently known) of a single formalaxiom that is compatible with all the known large cardinal axioms and that providesan axiomatic foundation for set theory that is immune to independence by Cohen’smethod. This axiom will not be unique, but there is the very real possibility that amongthese axioms there is an optimal one (from structural and philosophical considerations),in which case we will have returned, against all odds or reasonable expectation, to theview of truth for set theory that was present at the time when the investigation of settheory began.