In this three-part post I would like to motivate and provide a high-level overview of Woodin’s recent work in inner model theory, the goal being to describe the Ultimate- Conjecture, so that we might discuss the mathematics surrounding it and its potential philosophical significance. (All unattributed results below are due to Woodin.)
I will start in this first post by introducing the HOD Dichotomy Theorem. This dichotomy leads to a fork in the road, each side of which points to a different future of set theory. The first leads to the prospect of an ultimate inner model—one that is compatible with all (standard) large cardinal axioms—and the Ultimate- Conjecture is a precise conjecture as to what such an inner model might look like. The second leads to the prospect of a large cardinal hierarchy that transcends the axiom of choice. These two directions will be described in the second and third post, respectively.
Each possibility is of great foundational significance. In this respect, we are at a decisive phase in the development of the search for new axioms.
1. The HOD Dichotomy
As motivation for the HOD Dichotomy, we begin with the Dichotomy.
1.1. The Dichotomy
The following remarkable result is a combination of results due to Jensen, Kunen, and Silver, the most profound part being due to Jensen.
Theorem 1 (The Dichotomy Theorem) Exactly one of the following holds:
- For all singular cardinals ,
- is singular in , and
- Every uncountable cardinal is inaccessible in .