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Constructing regular ultrafilters from a model-theoretic point of view


Authors: M. Malliaris and S. Shelah
Journal: Trans. Amer. Math. Soc. 367 (2015), 8139-8173
MSC (2010): Primary 03C20, 03C45, 03E05
DOI: https://doi.org/10.1090/S0002-9947-2015-06303-X
Published electronically: February 18, 2015
MathSciNet review: 3391912
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Abstract: This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $ \mathrm {lcf}(\aleph _0, \mathcal {D})$ of $ \aleph _0$ modulo $ \mathcal {D}$, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters and thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, known to be detected by non-low theories. Assuming $ \kappa > \aleph _0$ is measurable, we construct a regular ultrafilter on $ \lambda \geq 2^\kappa $ which is flexible but not good, and which moreover has large $ \mathrm {lcf}(\aleph _0)$ but does not even saturate models of the random graph. This implies (a) that flexibility alone cannot characterize saturation of any theory, however (b) by separating flexibility from goodness, we remove a main obstacle to proving non-low does not imply maximal. Since flexible is precisely OK, this also shows that (c) from a set-theoretic point of view, consistently, OK need not imply good, addressing a problem from Dow (1985). Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for $ \mathcal {D}$ regular on $ \kappa $ and $ M$ a model of an unstable theory, $ M^\kappa /\mathcal {D}$ is not $ (2^\kappa )^+$-saturated; and for $ M$ a model of a non-simple theory and $ \lambda = \lambda ^{<\lambda }$, $ M^\lambda /\mathcal {D}$ is not $ \lambda ^{++}$-saturated. In the third part of the paper, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on $ SOP_2$. We prove that if $ \mathcal {D}$ is a $ \kappa $-complete ultrafilter on $ \kappa $, any ultrapower of a sufficiently saturated model of linear order will have no $ (\kappa , \kappa )$-cuts, and that if $ \mathcal {D}$ is also normal, it will have a $ (\kappa ^+, \kappa ^+)$-cut. We apply this to prove that for any $ n < \omega $, assuming the existence of $ n$ measurable cardinals below $ \lambda $, there is a regular ultrafilter $ D$ on $ \lambda $ such that any $ D$-ultrapower of a model of linear order will have $ n$ alternations of cuts, as defined below. Moreover, $ D$ will $ \lambda ^+$-saturate all stable theories but will not $ (2^{\kappa })^+$-saturate any unstable theory, where $ \kappa $ is the smallest measurable cardinal used in the construction.


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Additional Information

M. Malliaris
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: mem@math.uchicago.edu

S. Shelah
Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel – and – Department of Mathematics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/S0002-9947-2015-06303-X
Received by editor(s): March 22, 2012
Received by editor(s) in revised form: September 10, 2013
Published electronically: February 18, 2015
Additional Notes: The first author was partially supported by NSF grant DMS-1001666
The second author was partially supported by grants DMS-0600940 and 1101597 and by a Gödel fellowship. The second author was also partially supported by the Israel Science Foundation grant 710/07. This is paper 996 in the second author’s list of publications
Article copyright: © Copyright 2015 American Mathematical Society