  Selection of the preferred classical set of states in the process of
decoherence -- so important for cosmological considerations -- is discussed
with an emphasis on the role of information loss and entropy. {\it Persistence
of correlations} between the observables of two systems (for instance, a record
and a state of a system evolved from the initial conditions described by that
record) in the presence of the environment is used to define classical
behavior. From the viewpoint of an observer (or any system capable of
maintaining records) {\it predictability} is a measure of such persistence.
{\it Predictability sieve} -- a procedure which employs both the statistical
and algorithmic entropies to systematically explore all of the Hilbert space of
an open system in order to eliminate the majority of the unpredictable and
non-classical states and to locate the islands of predictability including the
preferred {\it pointer basis} is proposed. Predictably evolving states of
decohering systems along with the time-ordered sequences of records of their
evolution define the effectively classical branches of the universal
wavefunction in the context of the ``Many Worlds Interpretation". The relation
between the consistent histories approach and the preferred basis is
considered. It is demonstrated that histories of sequences of events
corresponding to projections onto the states of the pointer basis are
consistent.
