LARGE CARDINALS BEYOND CHOICE

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails.

IN SEARCH OF ULTIMATE-LTHE 19TH MIDRASHA MATHEMATICAE LECTURES

We give a fairly complete account which first shows that the solution to the inner model problem for one supercompact cardinal will yield an ultimate version ofLand then shows that the various current approaches to inner model theory must be fundamentally altered to provide that solution.

Descriptive inner model theory

The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture (MSC). One particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large.

Diagonal Prikry extensions

§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.Strong covering implies weak covering.In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include1. X is “0# exists”, Kx is L, Y is “strongly”.2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

Distinct iterable branches

§1. Introduction. The basic problem of inner model theory is how to construct mice satisfying hypotheses appreciably stronger than “there is a Woodin limit of Woodin cardinals”. We have a family of constructions, the Kc-constructions, which ought to produce such mice under the appropriate hypotheses on V. Perhaps the most important thing we lack is a proof that the countable elementary submodels of premice produced by a Kc-construction are ω1 + 1-iterable. The best partial results in this direction are those of Neeman ([4]) for Kc-constructions making use of full background extenders over V, and those of Andretta, Neeman, and Steel ([1]) for arbitrary Kc-constructions.Let be a countable premouse embedded by π into a level of the Kc-construction ℂ. If ℂ uses only full extenders over V as its background extenders, then π and ℂ enable one to lift an evolving iteration tree on to an iteration tree * on V. (See [3, §12].) The good behavior of * guarantees that of . The natural conjecture here is that V is ω1 + 1-iterable with respect to such trees* by the strategy of choosing the unique wellfounded branch. The open question here is uniqueness, since by [2] the uniqueness of the wellfounded branch chosen by * at limit stages strictly less than λ implies the existence of a wellfounded branch to be chosen at λ.