Coding and reshaping when there are no sharps

by Shelah and Stanley. [ShSt:294]
Set Theory Continuum, 1992
Assuming 0^sharp does not exist, kappa is an uncountable cardinal and for all cardinals lambda with kappa <= lambda < kappa^{+ omega}, 2^lambda = lambda^+, we present a ``mini-coding between kappa and kappa^{+ omega} . This allows us to prove that any subset of kappa^{+ omega} can be coded into a subset, W of kappa^+ which, further, ``reshapes the interval [kappa, kappa^+), i.e., for all kappa < delta < kappa^+, kappa = (card delta)^{L[W cap delta]} . We sketch two applications of this result, assuming 0^sharp does not exist. First, we point out that this shows that any set can be coded by a real, via a set forcing. The second application involves a notion of abstract condensation, due to Woodin. Our methods can be used to show that for any cardinal mu, condensation for mu holds in a generic extension by a set forcing.


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