At the end of last lecture, we showed Theorem 7, König’s lemma, stating that if for all then We begin by looking at some corollaries:
- If is a limit ordinal and is a strictly increasing sequence of nonzero cardinals, then
- If is an -indexed sequence of nonzero cardinals and for all then
- For any infinite , one has
Proof. 1. Notice that for all , so by König’s lemma. But, by an obvious injection,
2. Let If for all then König’s lemma gives us that
3. Use König’s lemma with , and for all Then
4. Let be a cofinal function. Let and for all Note that and
5. but, by 4.,
Note that once one decodes the argument, the proof of Cantor’s theorem above is exactly the same as the usual proof. On the other hand, Corollary 8.5 gives us more information and begins to show that the notion of cofinality is quite relevant in the study of cardinal arithmetic; much more dramatic illustrations of this claim are shown below. König’s lemma is the first genuinely new result in cardinal arithmetic (with choice) past Cantor’s theorem. For example, Specker’s result in Section I.5 and the Halbeisen-Shelah theorem in Section I.6 are both trivial once is assumed.
Some of the results that follow are available elsewhere in this blog, but I include them here to make these notes reasonably self-contained.
Definition. A cardinal is strong limit iff for all cardinals
Notice that the name is justified, since any strong limit is in particular ( or) a limit cardinal, i.e., an with a limit ordinal.
Of course, is a strong limit cardinal and, if holds below then is also strong limit. Without this assumption, examples of strong limit cardinals can be found by iterating the power set operation. More precisely, define the sequence of beth cardinals by and for a limit ordinal. Then the strong limit cardinals are precisely the with or a limit ordinal.
Theorem 9. (Bukovský-Hechler).
- If is infinite then
- In particular, if is strong limit, then
- Let be singular, and assume that the exponential is eventually constant below ; i.e., there is some and some such that for all cardinals with one has Then also
The theorem illustrates that the exponential map must satisfy certain restrictions, at least on its values on the singular cardinals. It is well known since the beginnings of forcing, from the work of Easton, that the exponential is `essentially’ arbitrary on the regular cardinals except that, of course, it is monotonic: If then and must obey König’s lemma, However, once large cardinals are considered, these are not the only restrictions. We will show this in Section 4.
Proof. 1. Let be infinite and let Let be strictly increasing, and let for all so Then as the obvious bijection verifies. But On the other hand, clearly for all so and therefore The result follows from these two inequalities.
2. If is strong limit and is as above, then On the other hand, for any so , and 2. follows from 1.
3. Assume is singular. With as above, if is sufficiently large, . But then , since we can take The result follows from 1.
For example, if for all (this situation is easily achieved by forcing) then also
The next result provides an easy `algorithm’ to compute any power, provided we know the values of the exponential function, the cofinality map, and the gimel function I’ll expand on this remark after the proof of the theorem.
Theorem 10. Let and be infinite cardinals. Let Then
Proof. 1. If then
2. If then any function is bounded, so and
3. Suppose that and is eventually constant (and equal to ) as approaches
Since is a singular cardinal and we can choose a strictly increasing sequence of cardinals cofinal in such that for all In particular, .
Note that for each so also [Of course, it follows from König’s lemma that in fact we have a strict inequality, but we only need this weaker estimate.]
We now have The other inequality is clear.
4. Finally, suppose that and is not eventually constant as approaches
Notice that if then Otherwise, for any , and the map would be eventually constant below after all. Hence,
Choose an increasing sequence of cardinals cofinal in Then The other inequality is clear.
Remark. By induction, it follows from Theorem 10 that the computation of the function `reduces’ to computations of the functions and for a cardinal, and the function for an ordinal. More precisely, if and are two models of set theory (with choice) with the same ordinals, and the values of any of these three functions are the same whether they are computed in or in , then also all the powers are the same, whether they are computed in or in
Forcing has shown that there is not much one can say about the exponential function when restricted to successor cardinals and small large cardinals. This indicates that to understand cardinal arithmetic, one’s efforts must concentrate on the exponential function on singular and large cardinals, and on the gimel function.
In fact, the gimel function suffices to compute the exponential by Theorem 9: First, for regular. Assume now that is singular. Let If is not eventually constant below then clearly , and by Theorem 9.1, while if is eventually constant below then by Theorem 9.3.
This observation explains why the bulk of research in cardinal arithmetic concentrates on understanding the gimel function. As we will see next time, this naturally leads to the study of certain infinite products. This latter study has proved quite fruitful. We will study two outcomes: Silver’s theorem, and one of the Galvin-Hajnal results. Then, we will briefly mention how Shelah has extended and generalized these theorems with his development of pcf theory.