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On the maximality of logics with approximations

Published online by Cambridge University Press:  12 March 2014

José Iovino
José Iovino*
Affiliation:
Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249-0664, USA, E-Mail: [email protected]
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Extract

In this paper we analyze some aspects of the question of using methods from model theory to study structures of functional analysis.

By a well known result of P. Lindström, one cannot extend the expressive power of first order logic and yet preserve its most outstanding model theoretic characteristics (e.g., compactness and the Löwenheim-Skolem theorem). However, one may consider extending the scope of first order in a different sense, specifically, by expanding the class of structures that are regarded as models (e.g., including Banach algebras or other structures of functional analysis), and ask whether the resulting extensions of first order model theory preserve some of its desirable characteristics.

A formal framework for the study of structures based on Banach spaces from the perspective of model theory was first introduced by C. W. Henson in [8] and [6]. Notions of syntax and semantics for these structures were defined, and it was shown that using them one obtains a model theoretic apparatus that satisfies many of the fundamental properties of first order model theory. For instance, one has compactness, Löwenheim-Skolem, and omitting types theorems. Further aspects of the theory, namely, the fundamentals of stability and forking, were first introduced in [10] and [9].

The classes of mathematical structures formally encompassed by this framework are normed linear spaces, possibly expanded with additional structure, e.g., operations, real-valued relations, and constants. This notion subsumes wide classes of structures from functional analysis. However, the restriction that the universe of a structure be a normed space is not necessary. (This restriction has a historical, rather than technical origin; specifically, the development of the theory was originally motivated by questions in Banach space geometry.) Analogous techniques can be applied if the universe is a metric space. Now, when the underlying metric topology is discrete, the resulting model theory coincides with first order model theory, so this logic extends first order in the sense described above. Furthermore, without any cost in the mathematical complexity, one can also work in multi-sorted contexts, so, for instance, one sort could be an operator algebra while another is. say, a metric space.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 4 , December 2001 , pp. 1909 - 1918
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Anderson, R. M., “Almost” implies “near”, Transactions of the American Mathematical Society, vol. 296 (1986), no. 1, pp. 229–237.Google Scholar
[2]Barwise, J. and Feferman, S. (editors), Model-theoretic logics, Springer-Verlag, New York, 1985.Google Scholar
[3]Fajardo, S. and Keisler, H. J., Neometric Forcing, preprint.Google Scholar
[4]Fajardo, S. and Keisler, H. J., Existence theorems in probability theory, Advances in Mathematics, vol. 120 (1996), no. 2, pp. 191–257.CrossRefGoogle Scholar
[5]Fajardo, S. and Keisler, H. J., Neometric spaces, Advances in Mathematics, vol. 118 (1996), no. 1, pp. 134–175.CrossRefGoogle Scholar
[6]Heinrich, S. and Henson, C. W., Banach space model theory. II. Isomorphic equivalence, Mathematische Nachrichten, vol. 125 (1986), pp. 301–317.CrossRefGoogle Scholar
[7]Henson, C. W., When do two Banach spaces have isometrically isomorphic nonstandard hulls?, Israel Journal of Mathematics, vol. 22 (1975), no. 1, pp. 57–67.CrossRefGoogle Scholar
[8]Henson, C. W., Nonstandard hulls of Banach spaces, Israel Journal of Mathematics, vol. 25 (1976), no. 1–2, pp. 108–144.CrossRefGoogle Scholar
[9]Iovino, J., Banach spaces and Banach space structures, II: Forking and compact topologies, Models, Algebras, and Proofs (Montenegro, C. and Caicedo, X., editors), Marcel Dekker, New York, 1998.Google Scholar
[10]Iovino, J., Stable Banach spaces and Banach space structures, I: Fundamentals, Models, Algebras, and Proofs (Montenegro, C. and Caicedo, X., editors), Marcel Dekker, New York, 1998.Google Scholar
[11]Keisler, H. J., From discrete to continuous time, Annals of Pure and Applied Logic, vol. 52 (1991), no. 1–2, pp. 99–141, Also, in International Symposium on Mathematical Logic and its Applications (Nagoya, 1988).CrossRefGoogle Scholar
[12]Lindström, P., On extensions of elementary logic, Theoria, vol. 44 (1969), pp. 1–11.CrossRefGoogle Scholar
[13]Lindström, P., A characterization of elementary logic, 1973, pp. 189–191.Google Scholar
[14]Shelah, S. and Stern, J., The Hanf number of the first order theory of Banach spaces, Transactions of the American Mathematical Society, vol. 244 (1978), pp. 147–171.CrossRefGoogle Scholar