What follows is not really part of the first section of [Sh:430], but it does pin down an important connection between cov and pp. The proposition is a special case of part of Shelah’s “cov vs. pp theorem”, Theorem 5.4 on page 87 of Cardinal Arithmetic. Note that there are no special assumptions on here. Getting inequalities in the reverse direction (i.e., showing that covering numbers are less than pp numbers) is generally a more difficult proposition.
Let us recall that is the minimum cardinality of a set such that for any (equivalently ), there is a such that .
Proposition 1 If is singular of cofinality , then
Proof: Suppose by way of contradiction that
By the definition of , we can find a (not necessarily increasing) sequence of regular cardinals and an ultrafilter on such that
Let be a family of cardinality standing as witness to , and let be a -increasing and cofinal sequence of functions in . For each , the range of is a subset of of cardinality at most , and so we can find such that
Since is a regular cardinal greater than , there is a single such that
Thus, (by passing to a subsequence of ) we may as well assume that the range of each is a subset of .
But , and so
by our choice of . Let us now define a function by setting if , and
whenever is in . Our assumptions imply that is in .
If , then for each we have
It follows that is a function in such that for all . This is absurd, given our choice of , and the proof is complete.
We’re going to be working with elementary submodels a lot, so I wanted to pin down the assumptions we make and the notation we use.
In the current project, we are investing combinatorics relevant to a singular cardinal , and is a regular cardinal “much much larger” than . The exact identity of isn’t important. We also need to fix some well-ordering of ; we will build this into the structures we consider so that they all have definable Skolem functions.
In general, denotes some expansion of by at most countably many functions, constants, and relations. I won’t get too pedantic with this, but for example, in the current situation we want to contain a name for , so that we know makes it into any elementary submodel of . We won’t make this explicit in our notation.
I will use (or just ) to denote the Skolem hull of in the structure .
I may update this post to reflect new things that I forgot to include…
Let us assume that is a singular cardinal with . We will prove that there is a family of cardinality such that every member of is a subset of some member of .
Fix a sufficiently large regular cardinal ; we will be working with elementary submodels of the structure where is some appropriate well-ordering of used to give us definable Skolem functions.
Let be a -approximating sequence, that is, is a -increasing and continuous sequence of elementary submodels of such that for each ,
- has cardinality ,
- is an initial segment of (so ), and
Let denote . Then is an elementary submodel of of cardinality containing as both an element and subset. Define
Clearly and , so we need only verify that for any , there is a with .
In broad terms, this will be done via an “ argument” with , but we’ll fill in the details in further posts.
We focus our attention on the first section of [Sh:430]: “equivalence of two covering properties”. The first result states the following:
Theorem 1 If , then .
I am pretty sure that this isn’t what Shelah meant to write here — the assumption that has uncountable cofinality means that the result is a trivial consequence of a deeper theorem in Cardinal Arithmetic [Update: Perhaps not! See 2nd update below]. I think that the proof he gives works even for the countable cofinality case, and this gives us something of interest because could very well be a fixed point. So, here’s what I conjecture he meant to say (after changing notation and translating the “cov” statement into more standard form):
Theorem 2 Let be a singular cardinal. Then if and only if there is a family of cardinality such that every member of is a subset of some element of .
We’ll work through his proof as best we can, and see if my conjecture is correct.
UPDATE 1: I don’t think the conjecture is correct. It looks to me like the proof was originally written for singular of countable cofinality, but a mistake was discovered and the statement was corrected but a lot of the old proof didn’t get revised properly. The annotated content was revised, but the abstract was not. Anyway, I’ll ask Saharon if I can’t figure out what’s going on.
UPDATE 2: I think the theorem doesn’t follow from the Cardinal Arithmetic stuff. The problem I run into is that when Shelah says informally that “cov = pp when the cardinal has uncountable cofinality”, there are lots of disclaimers hidden in the background. But this underscores why I think what I’m doing here is important — I want to pin down what exactly is known and what exactly is still open in this area.
So, I’ve decided what I want to try to accomplish with this blog: I want to use it as a tool for developing a reasonable exposition of more advanced pcf material. I consider this an experiment, as I have no idea how effective this will be, and it is entirely possible that I decide the entire endeavour is silly.
My first project will be to develop a treatment of the first section of [Sh:430] “Further Cardinal Arithmetic”; my intermediate goal is to concentrate on resuts related to the “cov vs. pp” problem.
It is safe to say that most of what I will be doing is presenting results of Shelah; if something is unattributed then consider it a result of Shelah (although I tend to try and reformulate things to make them more understandable to myself).
Tags: compact cardinal, SCH
This is really just an experiment to see if I can actually get latex2wp to work for me, but maybe it has mathematical interest as well. Anyway, it is well-known that Shelah’s pcf theory can be used to give a proof of Solovay’s result that the Singular Cardinals Hypothesis holds above a compact cardinal. What follows is a fairly short proof of a more general result; the argument consists of chaining together some ideas of Shelah taken from his Cardinal Arithmetic.
See “proof of 3.3 in Case gamma” and Remark 3.3B on pages 150-151 of his book.
Theorem 1 Suppose is singular of countable cofinality and . Then there is no uniform countably complete ultrafilter on .
Proof: We actually show more: we prove that there is no weakly -saturated uniform countably complete filter on . Said another way, if is a uniform countably complete filter on for singular of countable cofinality and , then we can find disjoint -positive subsets of . In particular, cannot be an ultrafilter.
To understand the proof, we don’t really need to know what means. What we really need is one of the combinatorial consequences of this statement, namely that there is a family of countable subsets of (not !) such that for every , the collection is essentially disjoint, in the sense that there is a function with domain such that
- is a finite subset of , and
- the collection is pairwise disjoint.
Thus, for each the collection can be “disjointified” by removing a finite subset from each .
Note that since the are subsets of , no -sized subcollection of can be disjointified, as this would give us disjoint subsets of . One should think of as a failure of “compactness” at which necessarily arises from the failure of the singular cardinals hypothesis at .
Let us assume now that is as above, and is a uniform countably complete filter on our cardinal ; we will show contains disjoint -positive subsets. Let us also assume that we have fixed some way of enumerating each , so that expressions such as “the first 5 elements of ” make sense.
Given and , define to be those greater than for which is contained in the first elements of . For each , the sequence is increasing with union . Since is uniform and countably complete, it follows that there is an such that is -positive.
For each , let be the “th element” of . Then clearly the set
is -positive for each , as includes .
We know for each , and so there exists an such that
Given , we define
We are now done, as given in , we see that , and therefore the collection is a collection of disjoint -positive subsets of .
Corollary 2 If where is strongly compact and is a singular cardinal of countable cofinality, then .
Proof: The co-bounded filter on is a uniform -complete filter. Since is strongly compact, this filter can be extended to a -complete ultrafilter (necessarily uniform) on . The conclusion now follows immediately from the theorem.
Corollary 3 If is strongly compact, then the Singular Cardinals Hypothesis holds above .