The Generalized Continuum Hypothesis revisited

Saharon Shelah

Published 1998-09-15Version 1

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, denoted lambda^{[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, (for all theta<mu)(2^theta<mu)), e.g. mu=beth_omega. Then for every lambda >= mu for some kappa<mu we have: (a) kappa <= theta < mu => lambda^{[theta]}= lambda and (b) there is a family 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 P .