In this post, I will work through the proofs of the following theorems. None of these reuslts are mine, and anything I don’t attribute is probably folklore.
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.
Lemma (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.
This will complete the proof of the Gaifman-Rowbottom theorem that we began in the last post. We will end up proving Gaifman’s theorem in a series of lemmas, of which Jensen’s lemma above is one. We begin by showing that, if we have a model that is -linearly iterable by , and we can elementarily embed into with being the image of some in , then we can also iterate the ultrapower construction over using . In order to save some typing, we will say that the pair is -linearly iterable if is -linearly iterable by and its images.
Lemma: Assume is -linearly. Let be such that is transitive, and there exists a elementary such that that . Then, is -linearly iterable.
Proof: The basic idea here is to show that one can iterate by iterating , and using the embedding at each stage to check that the iterates of are well-founded. In other words, setting , we want to create the following commutative diagram.
Luckily for us, the natural idea works here. We will let denote the typical element of the . At stage , we already have the map . So, we want to see where we need to map , but the obvious thing works here. That is, we set . So in general, for successors, we set .
At limit stages , we set equal to the map between direct limits we obtained from the uniqueness lemma in the previous post. More specifically, taking the direct limit of the models , we end up with some transitive (after taking the Mostowski collapse) . But, because we also have corresponding embeddings , we also see that the models elementarily embed into . Thus, taking the direct limit of the models , we end up with a model and an elementary embedding . This gives us that the diagram above commutes with each of the models transitive, and an elementary embedding . But then, each of the models are well-founded.
We will now prove Jensen’s lemma above.
Proof: The direction of this lemma follows directly from the special case of the shift lemma above. To prove the other direction, assume that is not linearly iterable, we will show that there is some countable transitive as in the statement of the lemma that is not -iterable.
Let witness that is not iterable, i.e. that the direct limit, is ill-founded. By reflection, we may assume that is a set. Now fix a large enough such that , and . Let be countable, transitive such that is elementary, and such that . But by elementarity, we see that
Now, as is a countable transitive model of (at least) , we see that the direct limit of is in fact ill-founded by absoluteness, and in fact that it is ill-founded at some countable ordinal . But then, we have elementary with , and not -linearly iterable.
The above lemma gives us a rather nice corollary when characterizing linear iterability.
Corollary: is linearly-iterable if and only if it is -linearly iterable
The $(\Rightarrow)$ direction follows from definition.
To prove the other direction, suppose that is -linearly iterable, and further suppose, by way of contradiction, that is not linearly-iterable. If is a proper class, pick large enough so that witnesses is -linearly iterable, but not linearly-iterable. That is, is -linearly iterable, but not linearly iterable. Let be the countable, transitive model that witnesses failure of linear-iterability of as in Jensen’s lemma, and fix witnessess with such that . But then, the shift lemma tells us that is at least -linearly iterable, since we picked large enough to witness that is -linearly iterable. Thus, we have have a contradiction, and -linear iterability implies linear-iterability for such models .
The nice part about the above corollary is that, if we have a failure of linear iterability, we know that it must have failed at some countable limit ordinal . We now move to a proof of Gaifman’s theorem.
Proof of Gaifman’s Theorem: Suppose otherwise, that is, suppose that has some ill-founded direct limit. Fix some large enough so that witnesses this, and in particular computes the functions that witness ill-foundedness correctly. Let be countable, transitive with such that there is a with . Let denote the “iterates” of , we want to show that each of these iterates is well-founded. We will do this by embedding each of them elementarily into , which is transitive.
In order to do this, we will construct maps elementary such that . In other words, we want to show that the following diagram commutes with each elementary:
That will then give us that each is well-founded, and therefore that is iterable giving us the desired contradiction. We will start with , and show how one constructs from there. It should be clear from the construction how to continue at successor stages. At limit stages, we just take the map defined in the “uniqueness” of direct limits lemma by expanding the above diagram to the following:
Now, with regards to constructing , let . Note that this intersection is countable, and therefore non-empty. Now define such that:
It is fairly routine to check that the above map is well-defined, and the initial segment of the above diagram commutes. That completes the proof.
I was planning on naively discussing mice for where is a real, but I feel like that would make this post too long. I don’t think I’ll end up talking about them much for now either, as I really should get started on working through the material in “The Stationary Tower”.