  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 .
