# 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. arXiv: math/9201249 DOI: 10.1007/978-1-4613-9754-0_21 MR: 1233827 -
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. - 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}, doi = {10.1007/978-1-4613-9754-0_21}, mrclass = {03E40 (03E10 03E45)}, mrnumber = {1233827}, mrreviewer = {James Baumgartner}, doi = {10.1007/978-1-4613-9754-0_21}, publisher = {Springer, New York}, note = {\href{https://arxiv.org/abs/math/9201249}{arXiv: math/9201249}}, arxiv_number = {math/9201249} }