Sh:1064

• Shelah, S. (2021). Atomic saturation of reduced powers. MLQ Math. Log. Q., 67(1), 18–42. arXiv: 1601.04824 MR: 4313125
• Abstract:
  2014.1.24 We continue [1019] looking at infinitary logics but now for filters instead ultrafilters. Earlier this month has written §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 §2 we deal with how to generalize “good ultrafilter” Intentions: (A) we hope to sort out the versions of cp implicit in §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?
• Version 2021-07-02 (39p) published version (25p)

@article{Sh:1064,
 author = {Shelah, Saharon},
 title = {{Atomic saturation of reduced powers}},
 journal = {MLQ Math. Log. Q.},
 fjournal = {MLQ. Mathematical Logic Quarterly},
 volume = {67},
 number = {1},
 year = {2021},
 pages = {18--42},
 issn = {0942-5616},
 mrnumber = {4313125},
 mrclass = {03C50 (03C20)},
 note = {\href{https://arxiv.org/abs/1601.04824}{arXiv: 1601.04824}},
 arxiv_number = {1601.04824}
}