# Sh:1064

- Shelah, S. (2021).
*Atomic saturation of reduced powers*. MLQ Math. Log. Q.,**67**(1), 18–42. arXiv: 1601.04824 DOI: 10.1002/malq.201900006 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.After publication a full proof was added of what was conclusion 3.10 in the published version and is here 2.10.

- Version 2023-06-12_3 (36p) published version (25p)

Bib entry

@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)}, doi = {10.1002/malq.201900006}, note = {\href{https://arxiv.org/abs/1601.04824}{arXiv: 1601.04824}}, arxiv_number = {1601.04824} }