I’ve had a chance to look over some of the posts for the second project, and I’ve been trying to fix typos and other infelicities, where by “infelicities” I mean “stupid things I say because I type without proofreading…”
Here we conclude a trio of posts dealing with elementary submodels and Skolem hulls. My goal in this post is to prove a slight strengthening of Observation 3.3A on page 379 of Cardinal Arithmetic, a result which will fill a critical role in the proof of the main theorem. Notice that what we are doing is just putting the two previous posts together into a single result.
Lemma 1 Suppose is a singular cardinal, let be as usual, and let be an elementary submodel such that . If there are objects and such that
- is a cardinal below ,
- , and
- for every ,
We see that the assumptions of Proposition 2 from yesterday’s post are satisfied if we define to be , and therefore the assumptions of Theorem 1 from the May 18, 2011 post are satisfied. If is not a subset of , then there is a regular such that
But then there is an such that
which is a contradiction as both and are in .
In this post, I want to give a more precise description of “Skolem hull arguments”, as such arguments are critical ingredients in many of the proofs we are looking at in this blog. I’ve done some searching, and I think I like the way Holst, Steffens, and Weitz present things in their book “Introduction to Cardinal Arithmetic” (referred to in the sequel as [HSW]), as they actually fill in several details that are usually glossed over and we’ll shortly be in need of this level of precision. I’m just going to sketch a little material from section 4.4 of their book, mainly to fix notation.
Our convention is that denotes a structure of the form , where is a well-ordering of . We’ll denote the usual language of set theory (i.e., one binary relation symbol standing for ) by , and let be together with another binary relation symbol standing for .
Given an formula in which occurs freely, let be the function from which sends each -tuple to the -least element for which
if there is such an , and which returns the value if there is no such . Thus, is a Skolem function for .
We usually think of the Skolem hull of as being the closure of under all Skolem functions, but the fact that our Skolem functions are definable in gives us the following:
Lemma 1 (Lemma 4.4.2 of [HSW]) If , then
The above will help make some of the arguments we present less technical than they might have been. Of course we will still be sloppy and refer to “elementary submodels of ”. One can take this as either “officially” referring to elementary substructures of , or referring to the -reducts of such structures.
For an easy example of a Skolem hull argument, let’s look again at the preceding post. Back then, we were considering elementary submodels and of , where was a regular cardinal much larger than some fixed singular cardinal . The theorem proved there needed two assumptions:
and now we can prove the following proposition:
Proposition 2 The two assumptions above are satisfied if
- is a cardinal below , and
Proof: Clearly only (4) is an issue, so assume . We know there is an formula , ordinals in , and in such that
Fix greater than , and define
The set definable in with parameters from , hence . Notice that lies in as well, as and . Since (and hence ) is a subset of , it follows that . Clearly , too, and so we have what we want.
Edit 5/27/2011 I left a piece of formula out of (7) —- there should be a “such that beta is the least such that…” in the definition of A. I’ll fix it when I get time.
This post should be considered a tentative one. What I’m trying to do is isolate a few facts that can be used to prove the various “elementary submodel results” scattered throughout the last chapter of Cardinal Arithmetic. I’ve already gone through several iterations of this, but the following result looks “stable” so I thought I’d put it up here.
Theorem 1 Suppose is a singular cardinal, and let be a sufficiently large regular cardinal. Suppose and are elementary submodels of such that
- , and
If , then there is a regular cardinal such that
Proof: Assuming , we define
Note the following facts:
Note that cannot be a successor ordinal: if it were, then its predecessor would witness the failure of (5). Since , we conclude
and so our proof will be complete if we can show that is in fact a regular cardinal.
Our proof hinges on the fact that is an element of as well. Why is this? By our assumptions, there is a set such that
Since , we conclude
and hence is definable from parameters available inside the model .
Thus, we have . If is not a regular cardinal, then
We can find such that maps onto a cofinal subset of . Since , it follows that there is an such that
But we know , and by (11), and therefore
In this post, we’ll lay out the notation and strategy for the proof.
Let us assume , , , and are cardinals with
whenever and satisfy
- is a progressive set of regular cardinals with and ,
- is a -complete ideal on extending the bounded ideal, and
- the structure has a true cofinality.
To save ourselves some time later, let us agree to call such and relevant.
Now let be a sufficiently large regular cardinal, and let be an elementary submodel of satisfying
- , and
- is an ordinal of cofinality greater than .
Our goal is to show that the collection
witnesses , i.e., that every set in can be covered by a union of fewer than sets from .
Now what about the “hard direction”? Given cardinals , , and such that is regular and , our aim is to prove
and moreover, if is regular, then there exist and such that
- is a progressive set of regular cardinals with and ,
- is a -complete ideal on extending the bounded ideal,
- the structure has a true cofinality, and
Notice that by the preceding post, condition 4 is equivalent to the statement
and so the equality
holds (at least in the case where ) in a strong sense. Note that the underlying issue here is that the pp number is defined as the supremum of a certain set of regular cardinals; we will show that if this supremum itself is a regular cardinal (and hence weakly inaccessible), then it actually occurs as the true cofinality of for some relevant and .
We’ll do this by proving the following:
Theorem 1 Suppose is a regular cardinal such that whenever and satisfy conditions 1, 2, and 3 above. Then as well.
The usual caveats apply here: it is closer to the truth to say that I will attempt to prove the above theorem. We’ll see what happens…
In this post, we’ll prove a more nuanced version of the result we discussed in our January 31 post. This is the “easy direction” of the theorem we’re aiming to prove, and the argument goes through without any special assumptions on the cofinality of the singular cardinal .
Theorem 1 Suppose , , and are cardinals satisfying . Then
Proof: Suppose is a progressive set of regular cardinals such that and . Further assume that is a -complete ideal on extending the bounded ideal such that has true cofinality, and let
We must show
Suppose this fails, and let witness . By our choice of , there is a sequence such that
- each is in ,
- , and
- if , then for some .
Since , it follows that for each the range of is covered by a union of fewer than sets from . Since is -complete, we conclude that for each there is a set such that
Now is regular and , so we can assume that there is a single set with the property that
for each .
Now define a function as follows:
Since , we know that is bounded in for all sufficiently large . Since contains the bounded subsets of , it follows that for each ,
But this is a contradiction, as there must exist an such that .