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June 6

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Why does the generalized continuum hypothesis implies the following equality for cardinal exponentiation?

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The article about the continuum hypothesis states that the the generalized continuum hypothesis implies the following:

when β+1 < α and , where cf is the cofinality operation.

How does one prove that?

Thanks!

Dan Gluck (talk) 12:59, 6 June 2020 (UTC)[reply]

See Talk:Continuum hypothesis#Proofs of certain consequences of GCH where I justify the claims in the article. Here I extract the relevant portion:
Since we are in the case that where cf is the cofinality operation, then any function from to must be bounded above by some
And γ has a cardinality where δ < α. :The cardinality of the set of functions so bounded by γ is
Adding these together for the possible values of γ gives
which means
OK? JRSpriggs (talk) 04:42, 7 June 2020 (UTC)[reply]