Independence, order, and the interaction of ultrafilters and theories
type: Item
instance of: scholarly article
title: Independence, order, and the interaction of ultrafilters and theories
zbMATH Open document ID: 1279.03060
published in: Annals of Pure and Applied Logic
publication date: 2012-09-26T00:00:00Z
reviewed by: J. M. Plotkin
Mathematics Subject Classification ID: 03C20
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C52
zbMATH DE Number: 6086896
zbMATH Keywords: unstable theories
zbMATH Keywords: regular ultrapowers
zbMATH Keywords: independence property
author: M. Malliaris
MaRDI profile type: Publication
full work available at URL: j.apal.2011.12.010
OpenAlex ID: W2067589138
cites work: On \(\vartriangleleft^{*}\)-maximality.
cites work: Ultraproducts which are not saturated
cites work: Ultraproducts in the theory of models
cites work: Ultrafilters and Independent Sets
cites work: Realization of \(\varphi \)-types and Keisler's order
cites work: The characteristic sequence of a first-order formula
cites work: Edge distribution and density in the characteristic sequence
cites work: Hypergraph sequences as a tool for saturation of ultrapowers
cites work: Homogeneous Universal Models.
cites work: There are just four second-order quantifiers
cites work: Classification theory and the number of non-isomorphic models.
cites work: Simple unstable theories
cites work: Q4284628
cites work: Toward classifying unstable theories
cites work: More on \(\mathrm{SOP}_1\) and \(\mathrm{SOP}_2\)
cites work: On sets of integers containing k elements in arithmetic progression
DOI: 10.1016/J.APAL.2011.12.010
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review text: The author considers the problem of realizing first-order types in regular ultrapowers. The author's previous work shows that it is enough to consider types determined by a single formula. In the article under review she investigates a class of formulas \(\varphi\) whose associated characteristic sequence of hypergraphs describes the realization of first-order and second-order types in ultrapowers, while at the same time pinpointing the properties of the corresponding ultrafilters.NEWLINENEWLINEThe author shows that each \(\varphi\) can be associated by means of its characteristic sequence to a ``second-order quantifier'' (in the sense of [\textit{S. Shelah}, Isr. J. Math. 15, 282--300 (1973; Zbl 0273.02009)]). The author goes on to show that many of Shelah's interpretability arguments go over to ultrapowers. As an application, she shows that any \(\varphi\) is dominated in Keisler's order by either the empty theory, the random graph, or by the minimal \(\mathrm{TP}_2\) theory. She also proves that the scope of the second-order quantifiers in Keisler's order does not go beyond \(\mathrm{TP}_2\). A final result indicates a possible gap in complexity between independence and strict order.
description: scientific article; zbMATH DE number 6086896