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KEISLER’S ORDER IS NOT LINEAR, ASSUMING A SUPERCOMPACT

Published online by Cambridge University Press:  01 August 2018

DOUGLAS ULRICH
DOUGLAS ULRICH*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF MARYLAND COLLEGE PARK, MD 20742, USAE-mail:ds_ulrich@hotmail.com
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Abstract

We show that if there is a supercompact cardinal, then Keisler’s order is not linear. More specifically, let Tn,k be the theory of the generic n-clique free k-ary graph for any n > k ≥ 3, and let TCas be the simple nonlow theory described by Casanovas in [2]. Then we show that TCas$$Tn,k always, and if there is a supercompact cardinal then Tn,k$$TCas.

Keywords

03C55uncountable model theoryKeisler’s orderultraproducts
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 2 , June 2018 , pp. 634 - 641
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Buechler, S., Lascar strong types in some simple theories, this Journal, vol. 64 (1999), no. 2, pp. 817–824.Google Scholar
Casanovas, E., The number of types in simple theories. Annals of Pure and Applied Logic, vol. 98 (1999), pp. 69–86.CrossRefGoogle Scholar
Keisler, H. J., Ultraproducts which are not saturated, this Journal, vol. 32 (1967), pp. 23–46.Google Scholar
Malliaris, M., Persistence and Regularity in Unstable Model Theory, Ph.D. thesis, University of California, Berkeley, 2009.Google Scholar
Malliaris, M., Realization of φ-types and Keisler’s order. Annals of Pure and Applied Logic, vol. 157 (2009), pp. 220–224.CrossRefGoogle Scholar
Malliaris, M. and Shelah, S., A dividing line within simple unstable theories. Advances in Mathematics, vol. 249 (2013), pp. 250–288.CrossRefGoogle Scholar
Malliaris, M. and Shelah, S., Constructing regular ultrafilters from a model-theoretic point of view. Transactions of the American Mathematical Society, vol. 367 (2015), pp. 8139–8173.CrossRefGoogle Scholar
Malliaris, M. and Shelah, S., Existence of optimal ultrafilters and the fundamental complexity of simple theories. Advances in Mathematics, vol. 290 (2016), pp. 614–681.CrossRefGoogle Scholar
Malliaris, M. and Shelah, S., Keisler’s order has infinitely many classes. Preprint, 2015, arXiv:1503.08341v2.Google Scholar
Shelah, S., Classification Theory, North-Holland, Amsterdam, 1978, Rev.. ed., 1990.Google Scholar
Ulrich, D., Low is a dividing line in Keisler’s order. Preprint, 2017, arXiv:1704.01537.Google Scholar