A maximality theorem for continuous first order theories

• Autorzy: Ackerman Nathanael , Karker Mary Leah

Identyfikatory

DOI

10.4467/20842589RM.22.005.16662

• Języki publikacji: EN

Abstrakty

EN

In this paper we prove a Lindstr¨om like theorem for the logic consisting of arbitrary Boolean combinations of first order sentences. Specifically we show the logic obtained by taking arbitrary, possibly infinite, Boolean combinations of first order sentences in countable languages is the unique maximal abstract logic which is closed under finitary Boolean operations, has occurrence number ω1, has the downward L¨owenheim-Skolem property to ω and the upward L¨owenheim-Skolem property to uncountability, and contains all complete first order theories in countable languages as sentences of the abstract logic. We will also show a similar result holds in the continuous logic framework of [5], i.e. we prove a Lindstr¨om like theorem for the abstract continuous logic consisting of Boolean combinations of first order closed conditions. Specifically we show the abstract continuous logic consisting of arbitrary Boolean combinations of closed conditions is the unique maximal abstract continuous logic which is closed under approximate isomorphisms on countable structures, is closed under finitary Boolean operations, has occurrence number ω1, has the downward L¨owenheim-Skolem property to ω, the upward L¨owenheim-Skolem property to uncountability and contains all first order theories in countable languages as sentences of the abstract logic.

Słowa kluczowe

EN

continuous logic; Lindström’s theorem; first order theory

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2022
• Tom: Vol. 57
• Strony: 61--93
• Opis fizyczny: Bibliogr. 11 poz., rys.

Twórcy

autor

Ackerman Nathanael

• Harvard University, Cambridge, MA 02138

autor

Karker Mary Leah

• Providence College, Providence, RI 02918

Bibliografia

• [1] N. L. Ackerman, Encoding complete metric structures by classical structures, Log. Univers. 14:4 (2020), 421–459.
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• [3] J. Barwise, Admissible sets and structures. An approach to definability theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin-New York, 1975.
• [4] I. B. Yaacov et al. Metric Scott analysis, Adv. Math. 318 (2017), 46–87.
• [5] I. B.Yaacov et al., Model theory for metric structures, in: Model theory with applications to algebra and analysis, Vol. 2. Vol. 350. London Math. Soc. Lecture Note Ser. Cambridge, Cambridge Univ. Press, 2008, pp. 315–427.
• [6] X. Caicedo, Maximality of Continuous Logic, in: Monographs and Research Notes in Mathematics (2017).
• [7] X. Caicedo and J. N. Iovino, Omitting uncountable types and the strength of [0,1]-valued logics, Ann. Pure Appl. Logic 165:6 (2014), 1169–1200.
• [8] C. Chang and H. J. Keisler, Continuous model theory. Annals of Mathematics Studies, No. 58. Princeton Univ. Press, Princeton, N.J., 1966.
• [9] C. W. Henson, Nonstandard hulls of Banach spaces, Israel J. Math. 25:1- 2 (1976), 108–144.
• [10] J. Iovino, On the maximality of logics with approximations, J.Symbolic Logic 66:4 (2001), 1909–1918.
• [11] P. Lindstr¨om, On extensions of elementary logic, Theoria 35 (1969), 1–11.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).