SCH,singular cardinal hypothesis, GCH, relationship

Question

Here it is stated that SCH is a consequence of GCH and also SCH is whether GCH might fail at a singular cardinal. These two condition seem yield a contradiction:does SCH follow from failing or holding GCH at singular cardinals?

Answer 1

$\mathsf{GCH}$ says that $2^\kappa=\kappa^+$ for all infinite cardinals $\kappa$. $\mathsf{SCH}$ says that $2^\kappa=\kappa^+$ for all singular strong limit cardinals $\kappa$. Clearly the first statement implies the second.

The question was, whether, if $\mathsf{GCH}$ is false, the smallest infinite cardinal $\kappa$ for which $2^\kappa>\kappa^+$ can be singular. Failure of $\mathsf{SCH}$ says that it can be, and $\mathsf{ZFC}+\neg\mathsf{SCH}$ is known to be equiconsistent with $\mathsf{ZFC}$ plus the existence of a sufficiently large measureable cardinal.