Proposition 1 Suppose is increasing in and is increasing in . Then
- , and
Here is where the choice of comes to our rescue. Since for each , it follows that
for each , and therefore
Now it follows easily that
Next, we define
Given our definition of , it is not hard to see
- , and
- every proper initial segment of is in .
Note that , as it is definable from , , and :
as it can be computed in by taking the Skolem hull of in the model .
General results on the approachability ideal tell us there is a sequence
- is a closed (not necessarily unbounded!) subset of ,
- has order-type at most , where ,
- , and
- the set of of cofinality with is stationary.
The above is a result from [Sh:420]; it says that there is a stationary subset of in the ideal . The details of this are worked out in my Handbook article — see Theorem 3.18 and Theorem 3.7.
Without loss of generality, the sequence lies in the model , hence in each and as well. Let be the -least continuous and increasing sequence of cardinals cofinal in satisfying ; note this means that the sequence is going to be in every elementary submodel of the structure .
Moving back to the main proof, let us fix ; recall that we want to show is a subset of some element , and the preceding post gives us our strategy.
For each and , we define
We note the following:
- , and
- both and have cardinality .
Next, we define
and let be the characteristic function of (note the “+”!) in , i.e., the domain of is , and for , we have
The function lies in , but there is no reason to believe it is an element of . Note, however, that IS an element of . It is also a progressive set of regular cardinals cofinal in , and therefore by earlier work, we know
is a scale in modulo the ideal of bounded subsets of .
Since is -directed modulo the ideal of bounded subsets of , it follows that for each , there exists a such that
The function mapping to need not be in . Nevertheless, the set of is closed unbounded in , and so we can find a of cofinality such that
- is closed under the map , and
- is closed unbounded in (of order-type ).
We’ll end this post on this cliff-hanger…
In this post, we examine another ingredient we’ll need in the course of the proof: the relationship between elementary submodels of and their “characteristic functions”. There are many variants of this lemma scattered throughout the book and allied literature, but we’ll just worry about what is needed for our current project. Remember that our conventions regarding elementary submodels were laid out in an earlier post.
Lemma 1 Let be elementary submodels of , and assume
- is unbounded in , and
- for any we have .
Proof: Suppose not. Define
Notice that exists by our assumptions, which also imply that cannot be a regular cardinal, as and
The proof will be complete if we manage to prove to the contrary that MUST be a regular cardinal.
Clearly must be a limit ordinal because is closed under the predecessor function. If is not a regular cardinal, then is a regular cardinal in (and ), and therefore contains a function mapping onto a cofinal subset of .
Since , it follows that and therefore . In the model , there must be an with . But since must be in as well, it follows that
contradicting the definition of .
How is this going to be used? Let us agree for a moment to call models and as above “-twins”. In the context of the main project we have an elementary submodel obtained as the union of a -approximating sequence . The goal is to show that any is a subset of some element of . We do this by producing an elementary submodel satisfying
- (so ), and
- and are -twins.
The conclusion of the lemma tells us that
which is exactly what we need.
Returning now to our main project, we extract a lemma from Shelah’s argument and examine it in isolation.
Let and be as assumed, and consider the structure
where denotes the ideal of bounded subsets of . It’s pretty easy to see that this structure is -directed, i.e., that every collection of fewer than functions in has a -upper bound.
Thus, we can define a sequence of functions in by letting be the -least function in serving as a -upper bound for . Using this definition, we achieve
for each , and so
Lemma 1 Let be a set of regular cardinals satisfying
- is progressive, and
Then is a scale in modulo the ideal of bounded subsets of .
Proof: Let satisfy the assumptions of the proposition. Since , we know that carries a scale of length modulo the ideal . Since is an elementary model of and , it follows that there is such a scale living in , say . Note that each is in as .
Now consider . Since is unbounded in , it follows easily that this sequence is increasing modulo in . Given , there is an with . But clearly this implies
Thus, the sequence is also cofinal in modulo , and we are done.
Notice that the above lemma gives us a way to smuggle information into : given as in the proposition, it follows that ANY function , whether in or not, is bounded modulo by a function of the form from . Formulating this as a corollary, we obtain:
Corollary 2 Let be a set of regular cardinals as in the preceding lemma. If , then there is an such that
The main result of this post is natural, but unsurprising. Oddly enough, though, I can’t find this exact version published anywhere. I mentioned something in this direction in my Handbook of Set Theory argument; I wish I had thought about it a bit more back then, because the proof is certainly easy enough. Anyway, the result is the following:
Theorem 1 The following statements are equivalent for a singular cardinal :
- If is a cofinal progressive set of regular cardinals, then carries a scale of length modulo the ideal of bounded subsets of .
We prove first that (2) implies (1), as this implication is almost immediate. Suppose is a progressive set of regular cardinals cofinal in and of cardinality . Our assumptions imply that there is a sequence of functions from such that
- , and
- if , then there is an such that .
If is any ultrafilter on disjoint to , then the two above statements continue to hold modulo , and hence has cofinality at most . But clearly this cofinality must be at least as extends the co-bounded filter on . The statement now follows from the definitions involved.
(see Claim 1.3(4) on page 4 of Cardinal Arithmetic), that is, given a collection of functions in of cardinality at most , there is a single function in above them all modulo the ideal .
By Conclusion 3.2 on page 62 of Cardinal Arithmetic, is equal to the supremum of all cardinals of the form , where is an ultrafilter on disjoint to . This latter cardinal is at most , and so it follows that
We may as well make a short post dealing with the definition of for singular; this will also let me test the latex2wp setup at home.
Definition 1 If is a singular cardinal, then is the supremum of all cardinals of the form
- is a set of regular cardinals cofinal in ,
- , and
- is an ultrafilter on extending the co-bounded filter.
This is not the “official” definition given as Definition 1.1 on page 41 of Cardinal Arithmetic, but it is equivalent. The official definition is designed with a nod towards generalization and refinement.
Note that the above definition is not the same as looking at
The reason is that in the definition of we do not require the ultrafilters under consideration to extend the co-bounded filter on , and this can make a difference if : you can potentially get a very large in by virtue of an ultrafilter concentrating on an initial segment of . In this situation, though, the fact that is in isn’t really anything to do with , and so the definition of is designed to ignore this phenomenon.