Blowing up the power of a singular cardinal of uncountable cofinality with collapses

Publication date: 31 October 2020

Abstract: The {em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if

k a p p a

is singular strong limit, then

2^{k a p p a} = k a p p a^{+}

. We prove that given a singular cardinal

k a p p a

of {em cofinality}

e t a

in the ground model, which is a limit of suitable large cardinals, and

e t a^{+} = a l e p h_{g a m m a}

, then there is a forcing extension which preserves cardinals and cofinalities up to and including

e t a

, such that

k a p p a

becomes

a l e p h_{g a m m a + e t a}

, and SCH fails at

k a p p a

. Furthermore, if

e t a

is not an

a l e p h

-fixed point, then in our model, SCH fails at

a l e p h_{e t a}

. Our large cardinal assumption is below the existence of a Woodin cardinal. In our model we also obtain a very good scale.

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