Talk:Singular cardinals hypothesis

Two versions

Why does the first version of SCH:

SCH1: For all singular strong limit cardinals κかっぱ, we have 2^κかっぱ = κかっぱ⁺.

imply the second version?

SCH2: For all cardinals κかっぱ satisfying 2^{cf(κかっぱ)} < κかっぱ, we have κかっぱ^{cf(κかっぱ)} = κかっぱ⁺

It is clear that the κかっぱ 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)