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 being a 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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