The XI International Workshop on Set Theory took place October 4-8, 2010. It was hosted by the CIRM, in Luminy, France. I am very glad I was invited, since it was a great experience: The Workshop has a tradition of excellence, and this time was no exception, with several very nice talks. I had the chance to give a talk (available here) and to interact with the other participants. There were two mini-courses, one by Ben Miller and one by Hugh Woodin. Ben has made the slides of his series available at his website.
What follows are my notes on Hugh’s talks. Needless to say, any mistakes are mine. Hugh’s talks took place on October 6, 7, and 8. Though the title of his mini-course was “Long extenders, iteration hypotheses, and ultimate L”, I think that “Ultimate L” reflects most closely the content. The talks were based on a tiny portion of a manuscript Hugh has been writing during the last few years, originally titled “Suitable extender sequences” and more recently, “Suitable extender models” which, unfortunately, is not currently publicly available.
The general theme is that appropriate extender models for supercompactness should provably be an ultimate version of the constructible universe . The results discussed during the talks aim at supporting this idea.
Let be supercompact. The basic problem that concerns us is whether there is an -like inner model with supercompact in .
Of course, the shape of the answer depends on what we mean by “-like”. There are several possible ways of making this nontrivial. Here, we only adopt the very general requirement that the supercompactness of in should “directly trace back” to its supercompactness in .
- We use to denote the set .
- An ultrafilter (or measure) on is fine iff for all we have .
- The ultrafilter is normal iff it is -complete and for all , if is regressive -ae (i.e., if ) then is constant -ae, i.e., there is an such that .
- is supercompact iff for all there is a normal fine measure on .
It is a standard result that is supercompact iff for all there is an elementary embedding with , , and (or, equivalently, ).
In fact, given such an embedding , we can define a normal fine on by
Conversely, given a normal fine ultrafilter on , the ultrapower embedding generated by is an example of such an embedding . Moreover, if is the ultrafilter on derived from as explained above, then .
Another characterization of supercompactness was found by Magidor, and it will play a key role in these lectures; in this reformulation, rather than the critical point, appears as the image of the critical points of the embeddings under consideration. This version seems ideally designed to be used as a guide in the construction of extender models for supercompactness, although recent results suggest that this is, in fact, a red herring.
The key notion we will be studying is the following:
Definition. is a weak extender model for ` is supercompact’ iff for all there is a normal fine on such that:
- , and
This definition couples the supercompactness of in directly with its supercompactness in . In the manuscript, that is a weak extender model for ` is supercompact’ is denoted by . Note that this is a weak notion indeed, in that we are not requiring that for some (long) sequence of extenders. The idea is to study basic properties of that follow from this notion, in the hopes of better understanding how such an model can actually be constructed.
For example, fineness of already implies that satisfies a version of covering: If and , then there is a with . But in fact a significantly stronger version of covering holds. To prove it, we first need to recall a nice result due to Solovay, who used it to show that holds above a supercompact.
Solovay’s Lemma. Let be regular. Then there is a set with the property that the function is injective on and, for any normal fine measure on , .
It follows from Solovay’s lemma that any such is equivalent to a measure on ordinals.
Proof. Let be a partition of into stationary sets.
(We could just as well use for any fixed . Recall that
and similarly for and .)
It is a well-known result of Solovay that such partitions exist.
Hugh actually gave a quick sketch of a crazy proof of this fact: Otherwise, attempting to produce such a partition ought to fail, and we can therefore obtain an easily definable -complete ultrafilter on . The definability in fact ensures that , contradiction. We will encounter a similar definable splitting argument in the third lecture.
Let consist of those such that, letting , we have , and
is stationary in .
Then is 1-1 on since, by definition, any can be reconstructed from and . All that needs arguing is that for any normal fine measure on . (This shows that to define -measure 1 sets, we only need a partition of into stationary sets.)
Let be the ultrapower embedding generated by , so
We need to verify that . First, note that . Letting , we then have that . Since
it follows that . Let . In , the partition into stationary sets. Let
The point is that .
To prove this, note first that and that is an -club of , since is continuous at points. Thus, for all we have and it follows that is stationary in . Hence .
Since , then . But , and this is an -club. It follows that no other can meet stationarily. So , and this completes the proof.
Solovay’s lemma suggests that perhaps it is possible to build models for supercompactness in a simpler way than anticipated, by using ultrafilters on ordinals to witness supercompactness.
Our key application of the lemma is the following (which, Hugh points out, could easily have been discovered right after Solovay’s lemma was established):
Corollary. Suppose is a weak extender model for ` is supercompact’. Suppose is a singular cardinal. Then:
- is singular in .
Note that item 1. is immediate from covering if , but a different argument is needed otherwise. Item 2. is a very -like property of . It is not clear to what extent there is a non-negligible (in some sense) class of cardinals for which computes their cofinality correctly.
Proof. This is immediate from Solovay’s lemma. Both 1. and 2. follow at once from:
If is regular in , then .
If is singular but regular in , then , but this is impossible since is singular.
If is singular but , then , contradicting that is singular.
It remains to establish . For this, we use Solovay’s lemma within .
Let be a normal fine ultrafilter on such that and . Note that such exists, even if is not a cardinal in : Just pick a larger regular cardinal in , and project the appropriate measure.
By Solovay’s lemma there is such that is 1-1 on . Suppose that . In , let be club, . Then since for the ultrapower embedding induced by . However, if , then while , by fineness. Contradiction.
It follows that if is supercompact in and in a forcing extension a -regular turns into singular while measures on all in lift (so, in particular, supercompactness of is preserved in the extension), then is no longer a cardinal in the extension.
We arrive at a key notion. Say that an inner model is universal iff (sufficiently) large cardinals relativize down to . The corollary seems to suggest that weak extender models for supercompactness ought to be universal, so solving the inner model problem for supercompactness essentially solves the problem for all large cardinals. In fact, we have:
Universality Theorem. Suppose is a weak extender model for ` is supercompact’. Suppose , is elementary, and . Then .
We will present the proof in the next lecture. In brief: Any extender that coheres with and has large critical point is in . To see why this is a universality result, notice for example that if in there is a proper class of -huge cardinals (for all ), then there is such a class in . Contrast this with the traditional situation in inner model theory, where inner models for a large cardinal notion do not capture any larger notions. (Similar results hold for rank into rank embeddings and larger, though some additional ideas are required here.)
In a sense, the universality theorem says that must be rigid. This is not literally true, but it is in the appropriate sense that there can be no sharps for :
Corollary. Suppose is an extender model for ` is supercompact’. Then there is no with .
Proof. Otherwise, is amenable to , by the universality theorem. But then , contradicting Kunen’s theorem.
(This is another -like feature that inherits.) Note the restriction to . This cannot be removed:
Example. Suppose is supercompact and is measurable. Let be a normal measure on , and let be the -th iterate of the ultrapower embedding . Then:
- is a weak extender model for ` is supercompact’.
- , so we cannot drop “” in the Corollary.
- Let where is the critical sequence ( for all ). Then where is the -th iterate of . It follows that is closed under -sequences. Since is a forcing extension of by small forcing (Prikry forcing), is also a weak extender model for ` is supercompact’, and clearly as well. Hence, “” cannot be dropped from the Corollary, even if we require some form of strong closure of .
We are now in the position to state a key dichotomy result, the proof of which will occupy us in the third lecture.
Definition. is extendible if for all there is with and .
Lemma. Assume is extendible. The following are equivalent:
- is a weak extender model for ` is supercompact’.
- There is a regular that is not measurable in
- There is a such that .
Note that this is indeed a dichotomy result: In the presence of extendible cardinals, either is very close to , or else it is very far.
Conjecture. If is extendible, then is an extender model for ` is supercompact’.
Let us close with a brief description of the proof of the Dichotomy Lemma. Note we already have that items 2. and 3. follow from 1. To prove , given , we consider the -club filter on , and try in to split into stationary sets in . Failure of this will give us that is measurable in . Assuming 2., this means we succeed, and we will use the stationary sets to verify that normal fine measures on are absorbed into . Then extendibility will give us a proper class of such , and item 1. follows.