Set theory is presently at a cross roads, where one is faced with two radically different possible futures.
This is first indicated by Woodin's HOD Dichotomy Theorem, an analogue of Jensen's Covering Lemma with HOD in place of L. 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 are two opposing research programs leading to opposite sides of the dichotomy. The first program is the program of inner model theory. In recent years Woodin has shown that if inner model theory can reach one supercompact cardinal then it "goes all the way", and he has formulated a precise conjecture - the Ultimate-L Conjecture - which, if true, would lead to a fine-structural inner model that can accomodate all of the standard large cardinals. This is the future where pattern prevails.
The second program 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. In recent work - joint with Bagaria and Woodin - the hierarchy of large cardinals beyond choice has been investigated. 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. Perhaps it is even consistent... The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where there can be no fine-structural understanding of the standard large cardinals. This is the future where chaos prevails.
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Philosophy of Mathematics Seminar Convenors: Daniel Isaacson and James Studd