580 -Cardinal arithmetic (6)

February 17, 2009

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 \kappa-inaccessibility, see Definition II.2.20 from last lecture.

Theorem 1. Let \kappa,\lambda be uncountable regular cardinals, and suppose that \lambda is \kappa-inaccessible. Let (\kappa_\alpha:\alpha<\kappa) be a sequence of cardinals such that \prod_{\alpha<\beta}\kappa_\alpha<\aleph_\lambda for all \beta<\kappa. Then also \prod_{\alpha<\kappa}\kappa_\alpha<\aleph_\lambda.

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 \kappa,\lambda are uncountable regular cardinals, and that \lambda is \kappa-inaccessible. Let \tau be a cardinal, and suppose that \tau^\sigma<\aleph_\lambda for all cardinals \sigma<\kappa. Then also \tau^\kappa<\aleph_\lambda.

Proof. Apply Theorem 1 with \kappa_\alpha=\tau for all \alpha<\kappa. {\sf QED}

Corollary 3. Suppose that \kappa,\lambda are uncountable regular cardinals, and that \lambda is \kappa-inaccessible. Let \tau be a cardinal of cofinality \kappa, and suppose that 2^\sigma<\aleph_\lambda for all cardinals \sigma<\tau. Then also 2^\tau<\aleph_\lambda.

Proof. Let (\tau_\alpha:\alpha<\kappa) be a sequence of cardinals smaller than \tau such that \tau=\sum_\alpha\tau_\alpha, and set \kappa_\alpha=2^{\tau_\alpha} for all \alpha<\kappa. Then \prod_{\alpha<\beta}\kappa_\alpha=2^{\sum_{\alpha<\beta}\tau_\alpha}<\aleph_\lambda for all \beta<\kappa, by assumption. By Theorem 1, \prod_{\alpha<\kappa}\kappa_\alpha=2^{\sum_\alpha\tau_\alpha}=2^\tau<\aleph_\lambda as well. {\sf QED}

Corollary 4. Let \kappa,\rho,\tau be cardinals, with \rho\ge2 and \kappa regular and uncountable. Suppose that \tau^\sigma<\aleph_{(\rho^\kappa)^+} for all cardinals \sigma<\kappa. Then also \tau^\kappa<\aleph_{(\rho^\kappa)^+}.

Proof. This follows directly from Corollary 2, since \lambda=(\rho^\kappa)^+ is regular and \kappa-inaccessible. {\sf QED}

Corollary 5. Let \rho,\tau be cardinals, with \rho\ge2 and \tau of uncountable cofinality \kappa. Suppose that 2^\sigma<\aleph_{(\rho^\kappa)^+} for all cardinals \sigma<\tau. Then also 2^\tau<\aleph_{(\rho^\kappa)^+}.

Proof. This follows directly from Corollary 3 with \lambda=(\rho^\kappa)^+. {\sf QED}

Corollary 6. Let \xi be an ordinal of uncountable cofinality, and suppose that 2^{\aleph_\alpha}<\aleph_{(|\xi|^{{\rm cf}(\xi)})^+} for all \alpha<\xi. Then also 2^{\aleph_\xi}<\aleph_{(|\xi|^{{\rm cf}(\xi)})^+}.

Proof. This follows from Corollary 5 with \rho=|\xi|, \tau=\aleph_\xi, and \kappa={\rm cf}(\xi). {\sf QED}

Corollary 7. Let \xi be an ordinal of uncountable cofinality, and suppose that \aleph_\alpha^\sigma<\aleph_{(|\xi|^{{\rm cf}(\xi)})^+} for all cardinals \sigma<{\rm cf}(\xi) and all \alpha<\xi. Then also \aleph_\xi^{{\rm cf}(\xi)}<\aleph_{(|\xi|^{{\rm cf}(\xi)})^+}.

Proof. This follows from Corollary 4: If \sigma<{\rm cf}(\xi), then \aleph_\xi^\sigma=\aleph_\xi\sup_{\alpha<\xi}\aleph_\alpha^\sigma, by Theorem II.1.10 from lecture II.2. But \xi<(|\xi|^{{\rm cf}(\xi)})^+, so both \aleph_\xi and \sup_{\alpha<\xi}\aleph_\alpha^\sigma are strictly smaller than \aleph_{(|\xi|^{{\rm cf}(\xi)})^+}. {\sf QED}

Corollary 8. If 2^{\aleph_\alpha}<\aleph_{(2^{\aleph_1})^+} for all \alpha<\omega_1, then also  2^{\aleph_{\omega_1}}<\aleph_{(2^{\aleph_1})^+}.

Proof. By Corollary 5. {\sf QED}

Corollary 9.  If \aleph_\alpha^{\aleph_0}<\aleph_{(2^{\aleph_1})^+} for all \alpha<\omega_1, then also  \aleph_{\omega_1}^{\aleph_1}<\aleph_{(2^{\aleph_1})^+}.

Proof. By Corollary 7. {\sf QED}

Notice that, as general as these results are, they do not provide us with a bound for the size of 2^\tau for \tau the first cardinal of uncountable cofinality that is a fixed point of the aleph sequence, \tau=\aleph_\tau, not even under the assumption that \tau is a strong limit cardinal. 

Read the rest of this entry »