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Cofinality spectrum theorems in model theory, set theory, and general topology


Authors: M. Malliaris and S. Shelah
Journal: J. Amer. Math. Soc. 29 (2016), 237-297
MSC (2010): Primary 03C20, 03C45, 03E17; Secondary 03E05
DOI: https://doi.org/10.1090/jams830
Published electronically: April 9, 2015
MathSciNet review: 3402699
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Abstract: We connect and solve two long-standing open problems in quite different areas: the model-theoretic question of whether $ SOP_2$ is maximal in Keisler's order, and the question from general topology/set theory of whether $ \mathfrak{p} = \mathfrak{t}$, the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using model-theoretic methods.


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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, TheHebrew University of Jerusalem, Jerusalem, 91904, Israel, and Department of Mathematics, Hill Center - Busch Campus, Rutgers, The State University of NewJersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/jams830
Received by editor(s): August 27, 2012
Received by editor(s) in revised form: January 14, 2014, and January 11, 2015
Published electronically: April 9, 2015
Additional Notes: The first author was partially supported by NSF grant DMS-1001666, by Shelah’s NSF grant DMS-1101597, by a Gödel research fellowship, and by a Sloan fellowship.
The second author was partially supported by the Israel Science Foundation grants 710/07 and 1053/11. This is paper 998 in Shelah’s list of publications.
Article copyright: © Copyright 2015 American Mathematical Society