The Generalized Continuum Hypothesis revisited

Abstract

We argue that we solved Hilbert&#39;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 .