Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-21T13:29:27.978Z Has data issue: false hasContentIssue false
  • English
  • Français

SEPARABLE MODELS OF RANDOMIZATIONS

Published online by Cambridge University Press:  22 December 2015

URI ANDREWS  and
H. JEROME KEISLER
URI ANDREWS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN-MADISON MADISON, WI, 53706, USAE-mail: andrews@math.wisc.eduURL: http://www.math.wisc.edu/∼andrews
H. JEROME KEISLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN-MADISON MADISON, WI, 53706, USAE-mail: keisler@math.wisc.eduURL: http://www.math.wisc.edu/∼keisler
Get access Rights & Permissions [Opens in a new window]

Abstract

Every complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomization theory. We show that the randomization theory has a prime model if and only if the first order theory has a prime model. And the randomization theory has the same number of separable homogeneous models as the first order theory has countable homogeneous models. We also show that when T has at most countably many countable models, each separable model of TR is uniquely characterized by a probability density function on the set of isomorphism types of countable models of T. This yields an analogue for randomizations of the results of Baldwin and Lachlan on countable models of ω1-categorical first order theories.

Keywords

Primary 03C1503B50Secondary 03C2003C40continuous model theorymetric structurerandomization
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1149 - 1181
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Baldwin, John and Lachlan, Alistair, On Strongly Minimal Sets, this Journal, vol. 36 (1971), pp. 79–96.Google Scholar
Ben Yaacov, Itaï, Continuous and Random Vapnik-Chervonenkis Classes. Israel Journal of Mathematics, vol. 173 (2009), pp. 309–333.CrossRefGoogle Scholar
Ben Yaacov, Itaï, Berenstein, Alexander, Ward Henson, C., and Usvyatsov, Alexander, Model Theory for Metric Structures, Model Theory with Applications to Algebra and Analysis, vol. 2, London Mathematical Society Lecture Note Series, vol. 350 (2008), pp. 315–427.Google Scholar
Ben Yaacov, Itaï and Jerome Keisler, H., Randomizations of Models as Metric Structures. Confluentes Mathematici, vol. 1 (2009), pp. 197–223.CrossRefGoogle Scholar
Ben Yaacov, Itaï and Usvyatsov, Alexander, Continuous first order logic and local stability. Transactions of the American Mathematical Society, vol. 362 (2010), pp. 5213–5259.CrossRefGoogle Scholar
Billingsley, Patrick, Convergence of Probability Measures, Wiley, New York, 1968.Google Scholar
Buechler, Steven A., Vaught’s Conjecture for Superstable Theories of Finite Rank. Annals of Pure and Applied Logic, vol. 155 (2008), pp. 135–172.CrossRefGoogle Scholar
Chang, C.C. and Jerome Keisler, H., Model Theory, Dover, New York, 2012.Google Scholar
Cutland, Nigel, Some Theories Having Countably Many Countable Models. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 105–110.CrossRefGoogle Scholar
Garavaglia, Steven, Decomposition of Totally Transcendental Modules, this Journal, vol. 45 (1980), pp. 155–164.Google Scholar
Kechris, Alexander, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
Jerome Keisler, H., Model Theory for Infinitary Logic, North-Holland, Amsterdam, 1971.Google Scholar
Jerome Keisler, H., Randomizing a Model. Advances in Mathematics, vol. 143 (1999), pp. 124–158.CrossRefGoogle Scholar
Jerome Keisler, H. and Morley, Michael D., On the Number of Homogeneous Models of a Given Power. Israel Journal of Mathematics, vol. 5 (1967), pp. 73–78.CrossRefGoogle Scholar
Maharam, Dorothy, On Homogeneous Measure Algebras. Proceedings of the National Academy of Sciences of the USA, vol. 69 (1942), pp. 143–160.Google Scholar
Mayer, Laura, Vaught’s Conjecture for o-Minimal Theories, this Journal, vol. 53 (1988), pp. 146–159.Google Scholar
Morley, Michael D. and Vaught, Robert L., Homogeneous Universal Models. Mathematica Scandinavica, vol. 11 (1962), pp. 37–57.CrossRefGoogle Scholar
Scott, Dana, Logic with Denumerably Long Formulas and Finite Strings of Quantifiers, The Theory of Models (Addison, J. et al. ., editors), North-Holland, Amsterdam, 1965, 329–341.Google Scholar
Shelah, Saharon, Harrington, Leo, and Makkai, Michael, A Proof of Vaught’s Conjecture for ω-stable Theories. Israel Journal of Mathematics, vol. 49 (1984), pp. 259–280.CrossRefGoogle Scholar
Steel, John R., On Vaught’s conjecture, Cabal Seminar 76–77, 193-208, Lecture Notes in Mathematics, vol. 689, Springer, 1978.Google Scholar
Vaught, Robert L., Denumerable Models of Complete Theories, Infinitistic Methods, Warsaw, 1961, pp. 303–321.Google Scholar