On a theorem of silver

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Abstract

A recent theorem of Silver, in its simplest form, states, that if ω < cf(k) < k and 2λ+ for all λ < k, then 2k=k+. Silver's proof employs Boolean-valued as well as nonstandard models of set theory. In the present note we give an elementary proof of Silver's theorem in its general form.

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The work of the first author was partially supported by NSF Grant GP-38026; the work of the second author was supported by NSF Grant GP-43841 and by a Fellowship from the Institute for Advanced Study.