5. PCF theory
To close the topic of cardinal arithmetic, this lecture is a summary introduction to Saharon Shelah’s pcf theory. Rather, it is just motivation to go and study other sources; there are many excellent references available, and I list some below. Here I just want to give you the barest of ideas of what the theory is about and what kinds of results one can achieve with it. All the results mentioned are due to Shelah unless otherwise noted. All the notions mentioned are due to Shelah as far as I know.
- Maxim Burke, Menachem Magidor, Shelah’s pcf theory and its applications, Annals of pure and applied logic, 50, (1990), 207–254.
- Thomas Jech, Singular cardinal problem: Shelah’s theorem on , Bulletin of the London Mathematical Society, 24, (1992), 127–139.
- Saharon Shelah, Cardinal arithmetic for skeptics, Bulletin of the American Mathematical Society, 26 (2), (1992), 197–210.
- Saharon Shelah, Cardinal arithmetic, Oxford University Press, (1994).
- Menachem Kojman, The ABC of pcf, unpublished notes, available (as of this writting) at his webpage.
- Uri Abraham, Menachem Magidor, Cardinal arithmetic, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., forthcoming.
- Todd Eisworth, Successors of singular cardinals, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., forthcoming.
Pcf came out of Shelah’s efforts to extend and improve the Galvin-Hajnal results on powers of singulars of uncountable cofinality (see lectures II.6 and II.7). His first success was an extension to cardinals of countable cofinality. For example:
- If is strong limit, then
- In fact, for any limit ordinal
- If is limit and then
Notice that this is an improvement of the Galvin-Hajnal bounds even if has uncountable cofinality. As impressive as this result may be, Shelah felt that it was somewhat lacking in that the bound was not “absolute” since can be arbitrarily large. The best known result of pcf theory remedies this issue:
Theorem 2 If is a limit ordinal and then
These results are obtained by a careful study of the set products for sets of cardinals The basic idea is the following: Let be cardinals, and let be the cofinality of i.e., the minimal size of a collection of subsets of of size such that any such subset is covered by one of the family:
Then clearly On the one hand, if is cofinal in then any subset of of size is in and this set has size at most On the other hand, obviously
Hence, to bound at least if reduces to understanding Note that if then trivially
Definition 3 Let be cardinals such that and either or else ( and ).
Let denote the least such that there is a family of size of subsets of each of size less than such that whenever and there is some with and
To understand these covering numbers, Shelah switches to the study of cofinalities of for different ideals so his results directly relate to the Galvin-Hajnal approach. The switch is justified in view of the following result (the vs theorem):
Definition 4 A partially ordered set has true cofinality iff there is a -increasing and cofinal sequence of length of elements of If this is the case, we write
The definition above is nontrivial. For example, where is ordered by iff and However, there is no increasing -sequence cofinal in Similarly, we can arrange that is singular, although this is impossible if has a true cofinality.
Definition 5 If then the pseudopower is the supremum of the true cofinalities of where is a set of at most regular cardinals below and unbounded in and varies among the ideals over extending the ideal of bounded sets such that exists.
denotes and if is a class of ideals, then is defined just as but with only ideals in being considered. If is an ideal (extending the ideal of bounded sets), then denotes where is the class of ideals extending
Suppose that is regular, and Then:
- Moreover, if is regular and larger than then for some and we have and
- In item 1, the inequality holds without needing to require that
- In item 1, if both sides of the equation are larger than then the sup is in fact a max.
For a proof, see Theorem 5.4 of Shelah’s book. Shelah notes the following conclusions:
Corollary 7 Suppose that is a singular cardinal of uncountable cofinality. Then
Lemma 8 Suppose that and either or Then
Corollary 9 Assume that Then
unless in which case or and for some in which case
If is a prime ideal (i.e., the set of complements of members of an ultrafilter), then is a total quasi-order on i.e., the collection of equivalence classes under the relation ( iff ), is linearly ordered under the relation iff In effect, in this case is just the ultrapower of by the ultrafilter dual to By ‘s lemma, this is a linearly ordered set.
This leads to the name pcf, which stands for possible cofinalities of ultraproducts of sets of regular cardinals.
1. A detour
Before we continue, a few words are in order on bounds for fixed points of the aleph function. For them, pcf theory does not apply directly, so different approaches are required. For example, the following extension of the ideas of Galvin-Hajnal that, combined with Theorem 6, provides bounds for powers for “many” even if they are fixed points.
Theorem 10 Suppose that is regular and is increasing and continuous, and let be a normal ideal on Suppose that for all Then
Definition 11 By induction, define the classes of cardinals:
- is the class of infinite cardinals.
- for limit.
For example, is the class of fixed points of the aleph sequence.
Definition 12 By induction on define the cardinals as follows:
- is defined by induction on as follows:
- If is limit,
- If is limit, define by induction on
- If is limit,
Note that the function is monotone (nonstrictly) increasing in and Also, for fixed
If is limit,
One then has, for example, the following bounds:
Theorem 13 Let denote the class of normal ideals.
- If then
- If is the -st member of and then is strictly smaller than the -st member of
- If and is not a limit of weakly inaccessible cardinals, then there are no weakly inaccessible cardinals in
Shelah found several additional results about bounds for powers of fixed points. However, there are inherent limitations to these results, as illustrated by the following:
Theorem 14 (Gitik) If holds, is a strong cardinal, and there are no inaccessibles above then there is a forcing extension that preserves cofinalities and where the following hold:
- is the first member of
- holds below
The understanding of cardinal arithmetic at fixed points is still severely more limited than at other cardinals.
Definition 15 Let be a (nonempty) set of cardinals.
- A cardinal is a possible cofinality of iff there is an ultrafilter over such that
- is the set of possible cofinalities of
- for all ultrafilters over that contain
Clearly, is a (not necessarily proper) ideal. By considering principal ultrafilters, One easily verifies that implies and that
More interesting results are obtained assuming that is not “too large” relative to its members.
It follows that (for as in the theorem) the sequence
is increasing and continuous. Note that if then so the displayed sequence only requires
It also follows that in the definition of one can restrict oneself to prime ideals, provided that
Corollary 17 With as in Theorem 16:
- has a largest element.
The number of ultrafilters over an infinite set is so the bound on the size of is nontrivial.
Proof: 1. If then Since the sequence of ideals is increasing and continuous, and all are contained in the bound follows.
2. If then so there is a least such that By continuity of the sequence of ideals it must be the case that so is regular and the largest member of
Stronger results are obtained when is an interval of regular cardinals:
It follows immediately that If this is clear. Otherwise, one can take in Theorem 18.
It also follows that, for example, if then is regular.
By Theorem 18, and therefore the bound follows from the following theorem:
Theorem 19 If is limit and then
In general, for a “short” interval of regular cardinals, The main open problem in the area is whether is even possible. It has been recently shown by and Er-rhaimini that Shelah’s proof cannot be improved to give the bound
More specifically, if is an interval of regular cardinals such that then the operator has the following properties for any
- If then there exists with such that
- has a maximal element.
- If is a singular cardinal of uncountable cofinality then there exists a club such that
Abstractly, properties 1–4 suffice to show that Essentially, the properties allow one to identify with a topological closure operator, and one can then argue directly about the corresponding topology. A height can be attached to these structures, and the Er-rhaimini- theorem is that for it is consistent that there are topological structures satisfying properties 1–4 of height arbitrarily large below
3. Kunen’s theorem revisited
Closely related to property 4 above is the following result about the existence of scales:
Zapletal used this result to obtain a nice proof of Kunen’s theorem that there are no embeddings His argument is as follows:
Suppose towards a contradiction that is elementary and that if is the first fixed point of past its critical point, then Fix as in Theorem 20 with Let be strictly -increasing and cofinal in Then is -increasing and cofinal in in the sense of Since is cofinal in then in is also -increasing and cofinal in
Now define a function with domain by Note that since is regular in and strictly larger than then so
However, for all and all so is not cofinal in after all, since bounds it. Contradiction.
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