Large Cardinals Beyond Choice

Abstract

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.

The cofinality of the least Berkeley cardinal and the extent of dependent choice

“…For our purposes, it is useful to recall the definition of Berkeley cardinals and some results from , concerning the least Berkeley cardinal and the failure of . For any transitive set M , let be the collection of all non‐trivial elementary embeddings .…”

“…for the proof). As remarked above, the following theorem is proved in : Theorem Suppose that δ 0 is the least Berkeley cardinal. Let .…”

“…Proof Sketch The main point is the following claim, proved in : Claim For any set y there exists a transitive set N such that and y is definable (without parameters) in N .…”

“…Therefore, it makes sense to take under consideration the possibility that in the context of , the large cardinal hierarchy actually extends beyond the limit established by Kunen in the presence of . In this perspective, large cardinal axioms even stronger than Reinhardt cardinals have been introduced by the authors of , i.e., the hierarchy of Berkeley cardinals . In fact, a Reinhardt cardinal turns out to be consistent relative to the least Berkeley cardinal (by rank‐reflection).…”

“…The results presented in this article are from , containing an introductory account of Berkeley cardinals also. For more on the “choiceless” large cardinals see .…”