This semester, I plan on trying to work my through Paul Larson’s “The Stationary Tower”. Before tackling that, I decided to just read through some of my notes on ultrapowers, extenders, and elementary embeddings in order to refresh my memory a little bit. While doing this, I came across the following result:
Theorem (Gaifman and Rowbottom):
If there exists a measurable cardinal, then is uncountable.
Now, we certainly know that this result holds (and more) in the presence of , which is a strictly weaker assumption. That is, “ exists” gives us that the uncountable cardinals (in ) form a closed unbounded class of indiscernibles for , which immediately gives us that thinks each of these uncountable cardinals is strongly inaccessible. From this it follows that, in particular, is countable. The proof that gives us this closed unbounded class of indiscernibles be found in Devlin’s “Constructibility” I believe.
However, what’s nice about the above theorem is that the proof is a little more straightforward, and it goes through the machinery of linear iterations. While developing this machinery, we end up proving a special case of the shift lemma, and utilize the technique of comparing iterations. Both of these, in more complicated forms, come up when working with mice, as does the notion of iterability. The idea here is that measurable cardinals allows us to get our hands dirty with many of the basic notions that are important to inner model theory, but without many of the (extremely) technical difficulties.
For those reasons, I want to work through the proof of the Gaifman-Rowbottom theorem in this post. We will begin with the basics of linear iterations, but I will mostly only sketch arguments or provide a references as needed. As usual with this sort of thing, we will be working in Kelley-Morse. Let’s begin by reviewing a few basic properties of measurable cardinals (the proof of the theorem below can be found in Kanamori’s “The Higher Infinite”).
Theorem: Let be a measurable cardinal, be the witnessing ultrafilter, and be the elementary embedding induced by . The following hold:
1) , , and .
3) , and so .
Note that conditions 2 and 3 motivate the notions of -supercompact and -strong cardinals respectively.
Let be a measurable cardinal, be the witnessing ultrafilter, and be the ultrapower embedding given by . The above theorem tells us that , so does not witness that is measurable in . By elementarity however, witnesses that is measurable in . That is “ is a -complete, non-principal ultrafilter over “. So, we can perform the ultrapower construction inside with respect to and . It turns out that this ultrapower is well-founded, since well-foundedness is absolute between inner models of ZFC (in fact ZF – powerset + collection), and thinks that the resulting ultrapower is well-founded. So, we may identify this ultrapower with its transitive collapse. Let’s denote this model by , by , and by . Let denote the ultrapower embedding , and the ultrapower embedding .
Again, we can repeat this construction inside , and end up with a model , and an ultrapower embedding . Continuing in this matter, we end up with sequences and of ultrapowers and their associated elementary embeddings with the following properties.
1) , , and is the induced ultrapower embedding from to .
2) For , is an elementary embedding, and .
3) For , , and is the induced ultrapower embedding.
At this point, we have a directed system of elementary embeddings, and so we may take the direct limit of this system (if it exists), and end up with a model and elementary embeddings . Provided that this model is well-founded, we can identify with its transitive collapse, and continue on in this manner, taking ultrapowers at successor stages and direct limits at limit stages, provided that each of the models involved is well-founded. As we noted above, we can rule out ill-foundedness occurring at successor stages, but the absoluteness of well-foundedness, which means we only need to make sure limit stages are well-founded. Another thing to note is that, in order to iterate the ultrapower construction through successor stages, we can work in some such that “ZFC – powerset + collection + is a -complete, non-principal ultrafilter on + exists” (collection turns out to be stronger than replacement without the powerset axiom). We will denote this theory by for the sake of brevity, and note that we may assume that and are a part of our language. We will make use of this later. At this point, we will note that not only do direct limits exist, but they are “unique” in some sense.
Theorem (Existence): Let and be a directed system of elementary embeddings. Then, there exists some model in the language of set theory and maps such that
1) Each .
2) If , then .
3) For each , there is some with .
That is, the direct limit of the above system exists.
We begin by constructing the model . We say that is a thread if it is a function in with domain for some such that for all ,we have that and . In other words, we may think of as a thread that runs through in manner that is compatible with the elementary embeddings we are given. Note that one value of determines the rest of the values of in its domain. We now define an equivalence relation on threads:
Now, since our system may be a system of (proper) class models, we may use Scott’s trick here. We now define a binary relation on the equivalence classes of threads as follows:
Setting equal to the set of threads, we see that our desired model is . In particular, note that even if is a proper class, the relation is still set-like. From this construction, we can see what we want our embeddings to look like. In particular we define by the mapping such that . That this construction satisfies the three desired properties follows from construction.
Theorem (Uniqueness): Let and be a directed system of elementary embeddings with direct limit . Let be the associated elementary embeddings. Furthermore, suppose that there is some model and elementary embeddings satisfying for . Then, there exists an elementary embedding such that .
Both of the above theorems are proved in the introduction of “The Higher Infinite” and are fairly routine. We will now isolate the notion of linear iterability.
Definition: Let be a measurable cardinal, and be the witnessing measure, and let . For an ordinal , we say that the sequences , , and is a linear iteration of by of length if the following hold:
1) , and form a directed system of elementary embeddings.
2) , for , , and if is a limit ordinal, then is the direct limit of the directed system , and .
3) For an ordinal , the map is the induced ultrapower embedding, and for ordinals with limit, the maps are the direct limit maps defined earlier.
4) For , .
5) For , each is well-founded, and has been identified with its transitive collapse.
We say that is -linearly iterable by and its images if there exists a linear iteration of by of length . We say that is linearly iterable by and its images if it is -linearly iterable for each ordinal .
At this point, I am going to state (without proof) the two theorems that we need to prove the Gaifman-Rowbottom theorem, and then work through the proof of that theorem. I want to delay the proofs regarding linear iterability to the next post because I don’t want this post to get too long, and this seems like a nice division (maybe). Plus, I want to mention something about mice for for a real , and a discussion of linear iterability seems like a good transition. We’ll see if that actually works out.
Theorem (Gaifman): Let be a cardinal such that there is a -complete, non-principal ultrafilter on , and let . Then, is linearly iterable by and its images.
Theorem (Jensen): Let and be as above. is linearly iterable if and only if, for every countable transitive with a such that there is an elementary with , is -iterable.
The proofs of the above theorems are actually rather nice, and use some fun machinery. It seems worth it to prove those two theorems in a seperate post. Now, we will prove the Gaifman-Rowbottom theorem.
Theorem (Gaifman and Rowbottom): If there exists a measurable cardinal, then is uncountable.
Let be measurable, and fix a witness to the measurability of . We begin by noting that, by the above results, is iterably by , and we may reflect down to some such that . Let be countable, transitive such that there is some elementary embedding and some with . Then, by Jensen’s result above, is -linearly iterable by , and it follows that the direct limit is well-founded, and we may identify it with its transitive collapse.
To see this, suppose otherwise, i.e. that there is some sequence of elements in . Fixing representatives of these equivalence classes, there is then some ordinal such that . But then we have that , which is a contradiction. Thus, we have a sequence of iterates of by of length . Next, note that the sequence is a length increasing sequence of ordinals and therefore that .
Next, recall that , and therefore we have that . Additionally, note that for any , and any , we have . However, this tells us that . But since is countable, we see that is countable as well.
Here, we see a nice application of the machinery of iterated ultrapowers. With that in mind, I want discuss iterations a bit more in the next post by actually proving Gaifman’s and Jensen’s theorems, and showing how we can generalize these tools. Also, my notes on iterated ultrapowers are based on notes by my Masters Thesis advisor, Andrés Caicedo at Boise State University. I’ll see if I can find a public copy of those notes, as they’re much better than my blog post explanations.