### The Generalized Continuum Hypothesis revisited

by Shelah. [Sh:460]

Israel J Math, 2000

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, (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 .

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