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 .
This is a remarkable dichotomy. In the first case (where is close to ) the cardinal structure of provides us with a good approximation to that of , while in the second case (where is far ) the cardinal structure of is a very poor approximation of that of , since every (truly) uncountable cardinal—, , etc.—is thought is misidentified as being an inaccessible cardinal (and much more—it is thought to be a Mahlo cardinal, a weakly compact cardinal, an indescribable cardinal and, more generally, it is thought to have any large cardinal property that is consistent with .)
There is a “switch”—a certain real number, —the existence of which determines which case holds: If does not exist then we are in the first case, where is close to , and if exists then we are in the second case, where is far from . But, although is lurking in the background, the dichotomy is stated without reference to it.
A natural question is whether there is a generalization of the Dichotomy Theorem, where is replaced by richer inner models. It turns out that there are generalizations where is replaced by richer “-like” fine-structural models . But the real import of our question is whether there is in some sense an “ultimate” generalization of the dichotomy. To render this question precise let us compare the notion of definability involved in with richer notions of definability.
1.2. The Inner Models and HOD
Recall that is defined by letting , , (for limit ordinals ) and , where is the set of all subsets of which are first-order definable over with parameters from . Thus, is obtained by iterating first-order definability (locally) along the stem of the ordinals.
Now, there is a hierarchy of definability that goes far beyond that involved in the above construction. Each level of definability in this hierarchy can be transcended by a “diagonalization”, leading to a richer notion of definability. Gödel asked whether there was a notion of definability that was in some sense “absolute” (or “ultimate”) in that it could not be transcended in this way. He noticed two things. First, he noticed that any notion of definability which does not render all of the ordinals definable can be transcended (as can be seen by considering the least ordinal which is not definable according to the notion); and so any such notion must include OD, the collection of sets that are ordinal definable. Second, he noticed that the notion of ordinal definability cannot be so transcended (since by reflection OD is ordinal definable). It is for this reason that proposed the notion of ordinal definability as a candidate for an “absolute” (or “ultimate”) notion of definability.
However, OD is not quite what we are looking for, since OD is not a model of ZFC. But if one lets HOD be the collection of sets that are hereditarily OD, then HOD is a model of ZFC.
There are many respects in which the inner models and HOD are at opposite extremes.
- is built up from below by iterating local, first-order definability along the ordinals, while HOD is built from above in a way that renders it entangled with (indeed HOD is simply where is the -theory of the ordinals.)
- If exists then is not a class-generic extension of (in fact, , a real number, is not in any class-generic extension of ), while as a theorem of ZFC (and so under any additional large cardinal axioms) due to Vopenka, is a class-generic extension of HOD.
- The statement is not compatible with modest large cardinal assumptions (more precisely, it is not compatible with anything from and beyond), while the statement is compatible with all of the (standard) large cardinal assumptions.
So it is natural to ask whether there is a generalization of the Dichotomy Theorem where is replaced by the much richer model HOD. It turns out that there is.
1.3. The HOD Dichotomy
To describe the HOD Dichotomy we first have to introduce a couple of notions.
Definition 2 A cardinal is an extendible cardinal if for all there is an elementary embedding
such that and .
Definition 3 An uncountable regular cardinal is -strongly measurable in HOD if there exists such that
- is a cardinal in HOD and , and
- There is no partition of the set
such that for each , is a stationary subset of .
Remark 1 It is routine to show that if is an uncountable regular cardinal which is -strongly measurable in HOD then is a measurable cardinal in HOD.
We can now state the HOD Dichotomy Theorem, due to Woodin.
Theorem 4 (HOD Dichotomy Theorem) Suppose that is an extendible cardinal. Then exactly one of the following holds:
- For all singular cardinals ,
- is singular in , and
- Every regular cardinal greater than or equal to is -strongly measurable in .
In the first case we say that is “close” to above and in the second case we say that is “far” from .
This is a remarkable dichotomy. In the first case, where is “close” to , the cardinal structure of provides us with a good approximation of that of , at least above the extendible , while in the second case, where is “far” , every regular cardinal greater than or equal to is thought in to be -strongly measurable (and hence inaccessible, Mahlo, weakly compact, indescribable and much more, such as measurable).
There are some key differences between the two dichotomy theorems. First, the HOD Dichotomy Theorem is not proved by cases on a “switch”, an analogue of for . Second, in the case of the HOD Dichotomy Theorem all of the action takes place above the extendible cardinal, that is, it is above the extendible cardinal that HOD is close to in the first case and far from in the second case.
1.4. Two Futures
It is natural to ask which side of the Dichotomy holds and which side of the HOD Dichotomy holds. In the case of the Dichotomy this is not something that can be settled in , unless in one could refute the existence of . And in the case of the HOD Dichotomy, we shall see, it cannot be settled in , unless one refutes the existence of a certain kind of large cardinal, a “choiceless cardinal”. But for present purposes the point is that we are asking a foundational question, one which (modulo an outright inconsistency) is tied up with the justification of new axioms, and, as such, any answer is going to have to be somewhat delicate.
Let us first deal with the Dichotomy. In this case there is good reason to believe that the second side of the dichotomy holds—that is “far” from —since there is good reason to believe that exists. And the same is true of the Dichotomies, for various fine-structural generalizations of . In these cases the future always fell on the second side. Let us now turn to the HOD Dichotomy.
Assume that there is an extendible cardinal . The HOD Dichotomy presents us with a fork in the road leading to two possible futures, one in which HOD is close to above , the other in which HOD is far from above . Which future holds?
Extrapolating from the case of and its generalizations one might think that for similar reasons the future lies on the second side of the HOD Dichotomy. But given the discussion above about the differences between HOD and L and its generalizations we should take pause. For we already know that there are many respects in which HOD is close to —for example, is a class-generic extension of HOD, and HOD can accomodate all (standard) large cardinals. Perhaps the closeness between HOD and is even greater, and the first side of the HOD Dichotomy maintains.
Let is introduce some terminology.
Definition 5 (The HOD Hypothesis) There is a proper class of cardinals such that is not -strongly measurable in HOD.
Definition 6 (The HOD Conjecture) The HOD Hypothesis is provable in ZFC.
Thus, the HOD Conjecture is the conjecture that (provably in ZFC) we are in the first half of the HOD Dichotomy (where HOD is close to ).
The HOD Conjecture is a surprising conjecture. It is not the sort of thing that one would readily conjecture. Indeed, when I was a graduate student it went by a different name—it was called the “Silly Conjecture”!
To see why it is such a surprising thing to conjecture, suppose that is a cardinal which is not -strongly measurable in HOD. Then, by definition, for all such that is a cardinal in HOD and , there is a partition of the set
such that for each , is a stationary subset of . In other words, we can effect in a definable fashion (that is, in HOD) a partition of the stationary set (defined in ) into stationary (in !) sets. That one might prove in that such definable partitions into truly stationary sets exist for arbitrarily large cardinals would be really quite astonishing. For this reason, on the face of it, it would be much more plausible to conjecture the opposite, in which case, by the HOD Dichotomy Theorem one would have the surprising result that HOD is close to above an extendible .
Let us now summarize the two futures:
- The first future is (in which HOD is close to above ) is given by the HOD Conjecture. The reasons for the HOD Conjecture come—as we shall see in the second post—from inner model theory. More precisely, the HOD Conjecture follows from the Ultimate- Conjecture and there is some inner model theoretic evidence for the Ultimate- Conjecture. In this sense the first future is the future in which “inner model theory wins” and where “pattern prevails”.
- The second future (in which HOD is far from above ) is given by the possibility of “choiceless large cardinals”, as we shall see in the third post. This is the future in which “chaos reigns”.
I will describe the first future and the reasons for the HOD Conjecture (aka the “Silly Conjecture”) in the next post.