# Sh:294

• Shelah, S., & Stanley, L. J. (1992). Coding and reshaping when there are no sharps. In Set theory of the continuum (Berkeley, CA, 1989), Vol. 26, Springer, New York, pp. 407–416.
• Abstract:
Assuming 0^\sharp does not exist, \kappa is an uncountable cardinal and for all cardinals \lambda with \kappa \leq \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.
• Version 1996-03-11_10 (10p) published version (10p)
Bib entry
@incollection{Sh:294,
author = {Shelah, Saharon and Stanley, Lee J.},
title = {{Coding and reshaping when there are no sharps}},
booktitle = {{Set theory of the continuum (Berkeley, CA, 1989)}},
series = {Math. Sci. Res. Inst. Publ.},
volume = {26},
year = {1992},
pages = {407--416},
publisher = {Springer, New York},
mrnumber = {1233827},
mrclass = {03E40 (03E10 03E45)},
doi = {10.1007/978-1-4613-9754-0_21},
note = {\href{https://arxiv.org/abs/math/9201249}{arXiv: math/9201249}},
arxiv_number = {math/9201249}
}