Talk:Singular cardinals hypothesis

Two versions
Why does the first version of SCH:
 * SCH1: For all singular strong limit cardinals &kappa;, we have 2&kappa; = &kappa;+.

imply the second version?
 * SCH2: For all cardinals &kappa; satisfying 2cf(&kappa;) < &kappa;, we have &kappa;cf(&kappa;) = &kappa;+

It is clear that the &kappa; in SCH2 must be singular, but not necessarily a strong limit. For example, which instance of SCH1 can be used to show that the configuration $$2^{\aleph_0}=\aleph_1$$, $$2^{\aleph_1}=\aleph_{\omega+5} = \aleph_\omega^{\omega}$$ is impossible?

--SCH12 (talk) 09:53, 18 August 2010 (UTC)