Notre Dame Journal of Formal Logic

Invariant Version of Cardinality Quantifiers in Superstable Theories

Alexander Berenstein and Ziv Shami

Abstract

We generalize Shelah's analysis of cardinality quantifiers for a superstable theory from Chapter V of Classification Theory and the Number of Nonisomorphic Models. We start with a set of bounds for the cardinality of each formula in some general invariant family of formulas in a superstable theory (in Classification Theory, a uniform family of formulas is considered) and find a set of derived bounds for all formulas. The set of derived bounds is sharp: up to a technical restriction, every model that satisfies the original bounds has a sufficiently saturated elementary extension that satisfies the original bounds and such that for each formula the set of its realizations in the extension has arbitrarily large cardinality below the corresponding derived bound of the formula.

Article information

Source
Notre Dame J. Formal Logic, Volume 47, Number 3 (2006), 343-351.

Dates
First available in Project Euclid: 17 November 2006

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1163775441

Digital Object Identifier
doi:10.1305/ndjfl/1163775441

Mathematical Reviews number (MathSciNet)
MR2264703

Zentralblatt MATH identifier
1113.03029

Subjects
Primary: 03C45: Classification theory, stability and related concepts [See also 03C48] 03C50: Models with special properties (saturated, rigid, etc.)

Keywords
cardinality quantifiers superstable theories

Citation

Berenstein, Alexander; Shami, Ziv. Invariant Version of Cardinality Quantifiers in Superstable Theories. Notre Dame J. Formal Logic 47 (2006), no. 3, 343--351. doi:10.1305/ndjfl/1163775441. https://projecteuclid.org/euclid.ndjfl/1163775441


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References

  • [1] Buechler, S., Essential Stability Theory, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1996.
  • [2] Pillay, A., Geometric Stability Theory, vol. 32 of Oxford Logic Guides, The Clarendon Press, New York, 1996.
  • [3] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, 2d edition, vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.