### Atomic saturation of reduced powers

by Shelah. [Sh:1064]

2014.1.24 We continue [1019] looking at infinitary logics but
now
for filters instead ultrafilters. Earlier this month has written
section 1
which generalize [Sh:c,VI,2.6], so do not depend on theta beinga compact cardinal, (this apply also to theta = {aleph_0} but

this is marginal here) In section 2 we deal with how to generalize
``good
ultrafilter'' Intentions:
(A) we hope to sort out the versions of cp implicit in section
2.
(B) Even assume T has theta-cp, can we sort out the maximal T
?
(C) Recall the problem of building a good filter of lambda such

that the quotient is a complete BA free enough failing chain
condition
2014.1.26 [here or F1396] Noted (Friday) that if a model is an
ultrapower by a normal ultrafilter on theta, then looking at all
sequences of theta formulas, we can colour by 2^theta
colours, such that the colour of a branch determine if the type is
realized.
THIS is one side on the other SIDE we start with a strong limit
mu of cofinality theta and a model of cardinality mu
which embed much, then build a directed system of sub-models, indexed
by subsets of lambda each of cardinality at most theta
(<)-directed, bounded below all members of a given stationary subset
of lambda of cofinality theta . Sort out.
2015-07-24 Add f1433 sec 1,2?

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