A Maximality Theorem for Continuous First Order Theories
Choose format
RIS BIB ENDNOTEA Maximality Theorem for Continuous First Order Theories
Publication date: 28.11.2022
Reports on Mathematical Logic, 2022, Number 57, pp. 61-93
https://doi.org/10.4467/20842589RM.22.005.16662Authors
A Maximality Theorem for Continuous First Order Theories
In this paper we prove a Lindström 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 Lüowenheim-Skolem property to ωand the upward Lüowenheim-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 Lindström 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 Lüowenheim-Skolem property toω, the upward Lüowenheim-Skolem property to uncountability and contains all first order theories in countable languages as sentences of the abstract logic.
[1] N. L. Ackerman, Encoding complete metric structures by classical structures, Log. Univers. 14:4 (2020), 421-459.
[2] J. Barwise and S. Feferman, eds. Model-theoretic logics. Perspectives in Mathematical Logic. Springer-Verlag, New York, 1985.
[3] J. Barwise, Admissible sets and structures. An approach to de_nability 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. Lindstrom, On extensions of elementary logic, Theoria 35 (1969), 1-11.
Information: Reports on Mathematical Logic, 2022, Number 57, pp. 61-93
Article type: Original article
Harvard University, Cambridge, USA
Providence College, Providence, RI 02918
Published at: 28.11.2022
Received at: 21.10.2020
Article status: Open
Licence: CC BY
Percentage share of authors:
Classification number:
Article corrections:
-Publication languages:
EnglishView count: 598
Number of downloads: 414