### 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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