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CONSTRUCTING MANY ATOMIC MODELS IN ℵ1

Published online by Cambridge University Press:  14 September 2016

JOHN T. BALDWIN
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO 851 S. MORGAN CHICAGO, IL60607, USAE-mail: jbaldwin@uic.eduURL: http://homepages.math.uic.edu/∼jbaldwin/
MICHAEL C. LASKOWSKI
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF MARYLAND COLLEGE PARK, MD20742-4015, USAE-mail: mcl@math.umd.eduURL: http://www.math.umd.edu/∼laskow/
SAHARON SHELAH
Affiliation:
HEBREW UNIVERSITY (AND RUTGERS UNIVERSITY) EINSTEIN INSTITUTE OF MATHEMATICS GIVAT RAM, JERUSALEM, 9190401, ISRAELE-mail: shelah@math.huji.ac.ilURL: http://shelah.logic.at/

Abstract

We introduce the notion of pseudoalgebraicity to study atomic models of first order theories (equivalently models of a complete sentence of ${L_{{\omega _1},\omega }}$). Theorem: Let T be any complete first-order theory in a countable language with an atomic model. If the pseudominimal types are not dense, then there are 20 pairwise nonisomorphic atomic models of T, each of size ℵ1.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Baldwin, John T., Categoricity, University Lecture Notes, vol. 51, American Mathematical Society, Providence, RI, 2009.Google Scholar
Baldwin, J. T. and Larson, Paul, Iterated elementary embeddings and the model theory of infinitary logic , Annals of Pure and Applied Logic, accepted.Google Scholar
Baldwin, J. T., Larson, P., and Shelah, S., Almost Galois ω-stable classes, this JOURNAL, vol. 80 (2015), pp. 763784. Shelah index 1003.Google Scholar
Farah, I., Ketchersid, R., Larson, P., and Magidor, M., Absoluteless for universally Baire sets and the uncountable II , Computational Prospects of Infinity Part II, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Singapore, 2008, pp. 163191.CrossRefGoogle Scholar
Farah, I. and Larson, P., Absoluteness for universally Baire sets and the uncountable I . Quaderni di Matematica, vol. 17 (2006), pp. 4792.Google Scholar
Hutchinson, J. E., Elementary extensions of countable models of set theory, this JOURNAL, vol. 41 (1976), pp. 139145.Google Scholar
Keisler, H. J., Model Theory for Infinitary Logic, North-Holland, Amsterdam, 1971.Google Scholar
Keisler, H. J. and Morley, M., Elementary extensions of models of set theory . Israel Journal of Mathematics, vol. 5 (1968), pp. 331348.Google Scholar
Larson, P., A uniqueness theorem for iterations, this JOURNAL, vol. 67 (2002), pp. 13441350.Google Scholar
Marker, D., Model Theory: An Introduction, Springer-Verlag, Berlin, 2002.Google Scholar
Shelah, S., Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in {L_{{\omega _1}\omega }}$ part A . Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 212240. Paper 87a.CrossRefGoogle Scholar
Shelah, S., Classification Theory and the Number of Nonisomorphic Models, second ed., North-Holland, Amsterdam, 1991.Google Scholar
Van der Waerden, B. L., Modern Algebra, Frederick Ungar Publishing, New York, 1949. First German edition 1930.Google Scholar