4. Strongly compact cardinals and
Definition 1 A cardinal is strongly compact iff it is uncountable, and any -complete filter (over any set ) can be extended to a -complete ultrafilter over
The notion of strong compactness has its origin in infinitary logic, and was formulated by Tarski as a natural generalization of the compactness of first order logic. Many distinct characterizations have been found.
Let’s begin by defining the (first order) infinitary logics
Definition 2 Let and be nonzero cardinals or Given a set of relation, function, and constant symbols (a language), the formulas of the language are defined inductively by closing under the following clauses:
- Atomic formulas are formulas.
- If is an formula, so is
- If is a set of fewer than many formulas, then is also an formula.
- If is a set of fewer than many variables, and is an formula, then so is
One can formalize the above even further by, for example, starting with variables for each ordinal -ary relation symbols for all and all ordinals and -ary function symbols for all and all ordinals and assigning a (well-founded) tree decomposition to each formula.
One then defines the corresponding semantics naturally. The well-foundedness of the tree associated to each formula is used to guarantee that the obvious inductive definition takes into account all formulas. For example, given a model for the language under consideration, an assignment of elements of to variables, a set of fewer than many variables, and an formula we say that holds iff there is a function such that if is the map that to assigns unless in which case then
One says that iff for all assignments
One defines free and bound variables as usual, and the standard notions carry through: A sentence is a formula without free variables, a theory is a collection of sentences, and a theory is satisfiable iff it has a model, i.e., a structure in the language of such that for all
Definition 3 An theory is -satisfiable iff whenever and then is satisfiable.
Hence the compactness theorem for first-order logic (i.e., for ) is the claim that if is an theory that is finitely (i.e., -) satisfiable, then itself is satisfiable.
A result of Keisler and Tarski shows that the corresponding notion of compactness for with is of large cardinal character in that it at least implies measurability of The argument naturally leads to the notion of a fine measure.
Definition 4 Given a cardinal and a set denote by the collection of subsets of of size strictly smaller than
An ultrafilter over is fine iff it is -complete, nonprincipal, and for any
If , say that is -compact iff it is uncountable and there is a fine ultrafilter over
Actually, the notion originally considered was not that of a fine measure, but a closely related one:
Definition 5 Let be cardinals. An ultrafilter over a set is -regular iff there is a family of elements of such that for every of size
Proof: Let be a fine ultrafilter over and let for all Then witnesses that is -regular.
Conversely, let be a -complete -regular ultrafilter as witnessed by the sequence Consider the projection given by Then is a fine ultrafilter over
Proof: Recall that for any nonprincipal ultrafilter its additivity the first such that is not -complete, is either or a measurable cardinal. Let be the filter of cobounded subsets of This is a -complete filter so, by assumption, it can be extended to a -complete nonprincipal ultrafilter over If is singular, the additivity of this ultrafilter cannot be but it is not smaller than and it is not larger than , so it would have to be But is not measurable, being a successor, contradiction.
- is strongly compact.
- is -compact for all
- is uncountable, and for all regular there is a uniform -complete ultrafilter over
- For all there is an elementary embedding with such that whenever and then there is some such that and
- For all there is an elementary embedding with such that and there is a set such that and
- is uncountable, and the compactness theorem holds for Any -satisfiable -theory is satisfiable.
- is uncountable, and for every there is a -complete, -regular ultrafilter.
Proof: 1 implies 2. Let The collection generates a -complete filter by closing under supersets (because is regular, by Lemma 7). By 1, there is a -complete ultrafilter over that extends it. This ultrafilter is clearly fine.
2 implies 5. We may assume that Let be a fine ultrafilter over and form the ultrapower embedding Obviously (by considering the sets for ), is not -complete. It follows that From, say, Theorem 6 and preceding remarks from last lecture, it follows that By choice, this implies that
The result follows, since we can take That simply means that for all That simply means that
5 implies 2. If and are as in 5, without loss, otherwise, replace with
Then is a fine ultrafilter over
2 implies 4. Again, take where is a fine measure over Given with say let be given by Then is as required.
4 implies 1. Let be a -complete filter over a set and let Take a witness of 4 for Let extend and have size smaller than in Then since in it is an intersection of fewer than members of the -complete filter Let be in this intersection, and define an ultrafilter by
Then is a -complete ultrafilter extending
(Notice we have shown that 4 and 5 are equivalent, although neither property seems to imply the other directly.)
2 implies 6. Let be a -satisfiable theory. We may assume that or there is nothing to show. For any let Let be a fine ultrafilter over and consider the ultraproduct
This is defined by a straightforward generalization of Definition 1 from lecture II.9: For set iff Let be the equivalence class of under this equivalence relation. Now we do not need Scott’s trick to make sense of since there are only set-many such functions to begin with. Let
be the collection of these equivalence classes. Define for an -ary relation symbol in the language of by
Define for an -ary function symbol in the language of by
Finally, define for a constant symbol in the language of by where for all
It is easy to check that these definitions make sense (i.e., they do not depend on the representatives of the equivalence classes under consideration), and they turn into a structure in the language of
The proof of ‘s lemma generalizes to this setting, since is -complete and, since is fine, this implies that for all
(Notice that the same argument gives a proof of the compactness of and shows that, over the existence of nonprincipal ultrafilters over any infinite set (an assumption strictly weaker than the axiom of choice) guarantees the compactness of )
6 implies 1. Let be a -complete filter over a set Consider the language that has a constant symbol for all a different constant symbol and a binary relation symbol Let be the theory in this language consisting of the union of the -theory of and The -completeness of ensures that is -satisfiable, and 6 implies that it is satisfiable. Let be a model of and define an ultrafilter over by
It is straightforward to check that is -complete and extends
(Notice that, over the same argument shows that compactness for implies (and is therefore equivalent to) the existence of nonprincipal ultrafilters over any infinite set.)
1 implies 3. Use 1 to extend the filter of cobounded subsets of to a -complete ultrafilter over and notice that it is necessarily uniform.
2 is equivalent to 7. By Lemma 6.
3 implies 7. This is a theorem of Ketonen. Notice first that is regular. By induction on one shows that if is regular then there is a -complete -regular ultrafilter over
Notice that the base of the induction () is clear.
Definition 9 Let be regular. An ultrafilter over is weakly normal iff whenever is regressive, there is some such that
First we check that we may strengthen the assumption on the ultrafilters given by 3:
The following notion will prove useful:
Definition 11 Given a -complete ultrafilter and an ordinal let
where, as usual, is the ultrapower embedding.
If is a -complete ultrafilter over and we say that is the first function of
Proof: Let be the first function of Then works.
It follows that for all regular there is a weakly normal uniform -complete ultrafilter over
It is straightforward to check that in Definition 12 is indeed an ultrafilter.
Assuming that is regular and exists, we claim that we can take (more carefully, this is an ultrafilter over but we obtain the desired ultrafilter over via a bijection.)
In effect, it is easy to check that this product is -complete and uniform. We need to verify that it is -regular. This follows from a general result.
Proof: Let be the first function of so Let
For each let be cofinal and of size smaller than and let be a function such that for all so is a cofinal subset of that, in has size smaller than
By induction define disjoint intervals for as follows: Given for some let and let be the least ordinal such that
Note that for all by construction, and therefore the sets are in for all However, if has size then
because all the have size smaller than and there are disjoint intervals for Hence, witnesses that is -regular.
Notice that Lemma 13 is in fact an equivalence. In order to make use of this result, we need to identify the first function for
Definition 14 Let be regular uncountable cardinals. Let be a weakly normal -complete uniform ultrafilter over For each let be a -complete ultrafilter over some set The -sum of the ultrafilters is the ultrafilter over given by
In particular, if for all the above reduces to the product
Definition 15 In the setting of Definition 14, let for all and let be the function
Lemma 16 In the setting of Definition 15,
In particular, if and for all then (up to a bijection), and is the first function of
Proof: By uniformity of Suppose that so
By definition of for each there is such that
By weak normality of there is a fixed such that But then
By a straightforward generalization of Lemma 13, we get:
Corollary 17 For as in Definition 14, is -regular iff iff is -regular.
In particular, is indeed -regular.
More generally, if the inductive assumption holds for all and is regular, we can choose to be a -complete -regular ultrafilter over some set and define to be the -sum of the ultrafilters Then is -regular, and we are done.
Corollary 18 Let be -compact for all where is strongly compact. Then is strongly compact.
Proof: By the local nature of the proof of the equivalence of conditions 2 and 3 in Theorem 8.
In terms of cardinal arithmetic, the goal of this lecture is to prove the following theorem of Solovay:
The proof below is due to Matteo Viale and uses the notion of a covering matrix, that has proved very useful as well in the presence of forcing axioms like or I have had occasion of using both the theory developed in the course of the proof of Theorem 8, and Viale’s machinery, in my own research.
Definition 20 Let be a cardinal. A precovering matrix for is an array of subsets of such that
We say that a precovering matrix is nontrivial iff there is some such that for all and The least such is denoted
Viale’s notion of a covering matrix satisfies several additional requirements. For our application, precovering suffices.
Lemma 21 Suppose that is a singular cardinal of cofinality Then there is a nontrivial precovering matrix for with
Proof: Let be a strictly increasing sequence of regular cardinals cofinal in and for each nonzero let be surjective. Define the entries of inductively by
Clearly this works. In fact, for any and all
Viale’s definition of the condition is more restrictive, but for our application, this more general version suffices. Notice that says something about cardinal arithmetic at at least if there are nontrivial precovering matrices for
for any precovering matrix for
Proof: This is immediate from the displayed equation in Definition 22.
Theorem 24 (Viale) Suppose that is strongly compact. Then holds for all regular
Proof: Let be regular, and let be a precovering matrix for Let be a -complete uniform ultrafilter over For each and each let
Since there is some such that for all
For each let
Since by the above, there is some such that
In particular, is unbounded in and we claim that witnesses the covering property for Let be countable. Then (by definition of ) for all so In particular, this set is nonempty. Let Then (by definition of ) and for all Hence,
Proof of Theorem 19: Let be strongly compact. We show that for all by induction on The result is clear if is regular, and follows by induction if by Silver’s theorem. Assume now that Let be a nontrivial precovering matrix for with
Strongly compact cardinals are a bit of an oddity in the large cardinal hierarchy in that they are of really high consistency strength (for example, if is -compact, then there are inner models of that contain all the reals; this is significantly above in consistency strength than, say, the existence of a proper class of strong cardinals). However, in terms of size they need not be too large; in fact, Magidor showed that it is consistent that the first strongly compact is also the first measurable cardinal.
In contrast with the situation from strong cardinals shown last lecture, the following is still open:
Question 25 (Woodin) Assume that is strongly compact and that holds below Does it follow that it holds everywhere?
For more on infinitary logic, see Jon Barwise, Solomon Feferman, eds., Model-theoretic logics, Springer (1985). Of particular interest is Part C (Infinitary languages), 269-441, and even more specifically, the two articles Mark Nadel, and admissible fragments, 271–316, and Max Dickmann, Large infinitary languages, 317–363.
For the general theory of strongly compact cardinals, see Akihiro Kanamori, The Higher Infinite, Springer (1994) and Jussi Ketonen, Strong compactness and other cardinal sins, Annals of Mathematical Logic 5 (1972), 47–76. For more on the covering property, see Matteo Viale, A family of covering properties, Mathematical Research Letters, 15 (2) (2008), 221–238.
Typeset using LaTeX2WP. Here is a printable version of (an almost final draft of) this post.