# Sh:460

• Shelah, S. (2000). The generalized continuum hypothesis revisited. Israel J. Math., 116, 285–321.
• Abstract:
We argue that we solved Hilbert’s first problem positively (after reformulating it just to avoid the known consistency results) and give some applications. Let \lambda to the revised power of \kappa be the minimal cardinality of a family of subsets of \lambda each of cardinality \kappa such that any other subset of \lambda of cardinality \kappa is included in the union of <\kappa members of the family. The main theorem says that almost always this revised power is equal to \lambda. Our main result is

The Revised GCH Theorem:

Assume we fix an uncountable strong limit cardinal \mu (i.e., \mu>\aleph_0, (\forall\theta<\mu)(2^\theta<\mu)), e.g. \mu= \beth_\omega. Then for every \lambda\geq\mu for some \kappa<\mu we have:

(a) \kappa\leq\theta<\mu\Rightarrow\lambda^{[\theta]}=\lambda and

(b) there is a family {\mathcal P} of \lambda subsets of \lambda each of cardinality <\mu such that every subset of \lambda of cardinality \mu is equal to the union of <\kappa members of {\mathcal P}.

• Version 2006-10-26_10 (44p) published version (37p)
Bib entry
@article{Sh:460,
author = {Shelah, Saharon},
title = {{The generalized continuum hypothesis revisited}},
journal = {Israel J. Math.},
fjournal = {Israel Journal of Mathematics},
volume = {116},
year = {2000},
pages = {285--321},
issn = {0021-2172},
mrnumber = {1759410},
mrclass = {03E50 (03E04 03E30)},
doi = {10.1007/BF02773223},
note = {\href{https://arxiv.org/abs/math/9809200}{arXiv: math/9809200}},
arxiv_number = {math/9809200}
}