3. The Galvin-Hajnal theorems.
In this section I want to present two theorems of Galvin and Hajnal that greatly generalize Silver’s theorem. I focus on a “pointwise” (or everywhere) result, that gives us information beyond the pointwise theorems from last lecture, like Corollary 23. Then I state a result where the hypotheses, as in Silver’s theorem, are required to hold stationarily rather than everywhere. From this result, the full version of Silver’s result can be recovered.
Both results appear in the paper Fred Galvin, András Hajnal, Inequalities for Cardinal Powers, The Annals of Mathematics, Second Series, 101 (3), (May, 1975), 491–498, available from JSTOR, that I will follow closely. For the notion of -inaccessibility, see Definition II.2.20 from last lecture.
Theorem 1. Let be uncountable regular cardinals, and suppose that is -inaccessible. Let be a sequence of cardinals such that for all Then also
The second theorem will be stated next lecture. Theorem 1 is a rather general result; here are some corollaries that illustrate its reach:
Corollary 2. Suppose that are uncountable regular cardinals, and that is -inaccessible. Let be a cardinal, and suppose that for all cardinals Then also
Proof. Apply Theorem 1 with for all
Corollary 3. Suppose that are uncountable regular cardinals, and that is -inaccessible. Let be a cardinal of cofinality and suppose that for all cardinals Then also
Proof. Let be a sequence of cardinals smaller than such that and set for all Then for all by assumption. By Theorem 1, as well.
Corollary 4. Let be cardinals, with and regular and uncountable. Suppose that for all cardinals Then also
Proof. This follows directly from Corollary 2, since is regular and -inaccessible.
Corollary 5. Let be cardinals, with and of uncountable cofinality Suppose that for all cardinals Then also
Proof. This follows directly from Corollary 3 with
Corollary 6. Let be an ordinal of uncountable cofinality, and suppose that for all Then also
Proof. This follows from Corollary 5 with and
Corollary 7. Let be an ordinal of uncountable cofinality, and suppose that for all cardinals and all Then also
Proof. This follows from Corollary 4: If , then by Theorem II.1.10 from lecture II.2. But so both and are strictly smaller than
Corollary 8. If for all then also
Proof. By Corollary 5.
Corollary 9. If for all then also
Proof. By Corollary 7.
Notice that, as general as these results are, they do not provide us with a bound for the size of for the first cardinal of uncountable cofinality that is a fixed point of the aleph sequence, not even under the assumption that is a strong limit cardinal.
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