### Localizations of infinite subsets of $\omega$

by Roslanowski and Shelah. [RoSh:501]

Archive for Math Logic, 1996

In the present paper we are interested in properties of forcing
notions which measure in a sense the distance between the ground
model reals and the reals in the extension. We look at the ways the
``new'' reals can be aproximated by ``old'' reals. We consider
localizations for infinite subsets of omega . Though each member
of [omega]^omega can be identified with its increasing
enumeration, the (standard) localizations of the enumeration does
not provide satisfactory information on successive points of the
set. They give us ``candidates'' for the n-th point of the set but
the same candidates can appear several times for distinct n . That
led to a suggestion that we should consider disjoint subsets of
omega as sets of ``candidates'' for successive points of the
localized set. We have two possibilities. Either we can demand that
each set from the localization contains a limited number of members
of the localized set or we can postulate that each intersection of
that kind is large. Localizations of this kind are studied in
section 1. In the second section we investigate localizations of
infinite subsets of omega by sets of integers from the ground
model. These localizations might be thought as localizations by
partitions of omega into successive intervals.

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