In the last entry, we worked through the proof of the first Galvin-Hajnal formula.
Theorem 1 (Galvin-Hajnal): Let and be uncountable regular cardinals such that, for all , we have that . Assume further that is a sequence of cardinals such that for all , the product . It then follows that .
In this entry, I want to discuss a particular astonishing corollary to the above theorem, and use this to talk a bit about the singular cardinal problem.
Corollary: If is a strong limit cardinal, then .
To see this, we begin by noting that, for every , we have that . Now let for , and note that for every
since is countable and is strong limit. Thus, the sequence satisfies the hypotheses of the above theorem with and . It follows that .
We can generalize the above argument to get the following theorem.
Theorem 2 (Galvin-Hajnal): Let be a singular strong limit cardinal of uncountable cofinality. Then, .
Of course, this says absolutely nothing about fixed points of the function. Furthermore, we saw in the proof of the first Galvin-Hajnal formula that we needed uncountability in order to ensure the order was well-founded, thus enabling to pick a minimal witness contradicting the conclusion of the theorem. Shelah, using his powerful pcf theory, was able to show that Theorem 2 holds for cardinals of countable cofinality, and that we can in fact replace with , the fourth cardinal successor of . This is in stark contrast to the situation for regular cardinals.
Theorem 3 (Easton): Let be a function from regular cardinals to cardinals with the following properties:
- If , then ;
Then, there exists a model of ZFC in which for each regular cardinal .
In other words, the Zermelo-König theorem is the only provable result in ZFC regarding the behavior of the continuum function for regular cardinals. However, we have just shown that this is far from the case for singular cardinals that are not fixed points of the function. The next natural question is whether or not we can provide any natural bounds for the continuum function for fixed points for the function. In general we cannot, but this requires large cardinals to show.
The simplest way to go about this is to assume we have a sequence of strong cardinals with limit . From here, we use the the existence of canonical normal measures along with a very nice ordering afforded by the existence of extenders to add -many Prikry sequences to . Unfortunately, this construction does not preserve cardinals. In order to prevent cardinals from collapsing, the usual Cohen forcing to blow up to is weaved into each of the blocks that make up the extender-based Prikry forcing. Unfortunately, one cannot simply use this forcing, and then try to collapse to a smaller fixed point. The problem here is that the elements of the Prikry sequences are indiscernibles, and so attempting to collapse will actually collapse all cardinals between and .
Shelah has shown that there are provable (in ZFC) bounds for the continuum function certain at smaller fixed points. However, he was unable to show that there exists a bound for the first fixed point. Gitik was able to show that there was in fact no bound for the first fixed point, but the construction is beyond me at the moment.
This is just one direction that we can take when asking about singular cardinals. For example, we can note that the continuum function on singular cardinals is incredibly sensitive to what happens below said singular cardinals. On the other hand, Shelah introduced a number of higher cardinal invariants that are much less sensitive to the behavior of cardinal arithmetic below the singular cardinal we are concerned with. There are a number of questions related to the relationships between these cardinal invariants that turn out to be fairly interesting. In particular, questions about singular cardinals will involve pcf theory, large cardinals, inner model theory, forcing, and infinitary combinatorics (stationary reflection, square principles, coloring theorems, etc.).
In the next entry, I plan on working through the proof of the second Galvin-Hajnal formula. After that, I want to further pursue the thread regarding blowing up powers of singulars using large cardinals. That’s the plan at least.