Talk:Singular cardinals hypothesis

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Numbers

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Two versions

Why does the first version of SCH:

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

imply the second version?

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

It is clear that the κ 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)[reply]