Abstract
We show how to encode, by classical structures, both the objects and the morphisms of the category of complete metric spaces and uniformly continuous maps. The result is a category of, what we call, cognate metric spaces and cognate maps. We show this category relativizes to all models of set theory (unlike the category of complete metric spaces and uniformly continuous maps). We extend this encoding to an encoding of complete metric structures by classical structures. This provide us with a general technique for translating results about infinitary logic on classical structures to the setting of infinitary continuous logic on continuous structures. Our encoding will also allow us to talk about not only the relations between complete metric structures, but also the potential relations between complete metric structures, i.e. those which are satisfied in some larger model of set theory. For example we will show that given any two complete metric structures we can determine if they are potentially isomorphic by looking at any admissible set which contains them both.
Similar content being viewed by others
References
Ackerman, N.L.: On transferring model theoretic theorems of \(\cal{L}_{\infty ,\omega }\) in the category of sets to a fixed Grothendieck topos. Log. Univers. 8(3–4), 345–391 (2014). https://doi.org/10.1007/s11787-014-0105-5. https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/s11787-014-0105-5. ISSN: 1661-8297
Ackerman, N.L.: Relativized Grothendieck topoi. Ann. Pure Appl. Log. 161(10), 1299–1312 (2010). https://doi.org/10.1016/j.apal.2010.04.003. ISSN: 0168-0072
Ackerman, N.L.: The number of countable models in categories of sheaves. In: Models, Logics, and Higher-Dimensional Categories, Vol. 53, pp. 1–27. CRM Proceedings and Lecture Notes, American Mathematical Society, Providence, RI (2011)
Ackerman, N.L.: The number of countable models in realizability toposes. J. Pure Appl. Algebra 216(8–9), 1994–2013 (2012). https://doi.org/10.1016/j.jpaa.2012.02.037. ISSN: 0022-4049
Barwise, J.: Admissible Sets and Structures, An Approach to Definability Theory, Perspectives in Mathematical Logic. Springer, Berlin (1975)
Becker, H., Kechris, A.S.: The descriptive set theory of Polish group actions, Vol. 232, pp. xii+136. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (1996). https://doi.org/10.1017/CBO9780511735264. ISBN: 0-521-57605-9
Ben Yaacov, I.: Definability of groups in \(\aleph _0\)-stable metric structures. J. Symb. Log. 75(3), 817–840 (2010). https://doi.org/10.2178/jsl/1278682202. ISSN: 0022-4812
Ben Yaacov, I.: On perturbations of continuous structures. J. Math. Log. 8(2), 225–249 (2008). https://doi.org/10.1142/S0219061308000762. https://doi-org.ezp-prod1.hul.harvard.edu/10.1142/S0219061308000762. ISSN:0219-0613
Ben Yaacov, I.: Stability and stable groups in continuous logic. J. Symb. Log. 75(3), 1111–1136 (2010). https://doi.org/10.2178/jsl/1278682220. ISSN: 0022-4812
Ben Yaacov, I., Berenstein, A., Henson, C.W., Usvyatsov, A.: Model theory for metric structures. In: Model Theory with Applications to Algebra and Analysis, Vol. 2. Vol. 350, pp. 315–427. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (2008). https://doi.org/10.1017/CBO9780511735219.011
Ben Yaacov, I., Iovino, J.: Model theoretic forcing in analysis. Ann. Pure Appl. Log. 158(3), 163–174 (2009). https://doi.org/10.1016/j.apal.2007.10.011. ISSN: 0168-0072
Ben Yaacov, I., Nies, A., Tsankov, T.: A Lopez–Escobar Theorem for Continuous Logic (2014). arXiv:1407.7102
Ben Yaacov, I., Usvyatsov, A.: Continuous first order logic and local stability. Trans. Am. Math. Soc. 362(10), 5213–5259 (2010). https://doi.org/10.1090/S0002-9947-10-04837-3. ISSN: 0002-9947
Ben-Yaacov, I.: Positive model theory and compact abstract theories. J. Math. Log. 3(1), 85–118 (2003). https://doi.org/10.1142/S0219061303000212. ISSN: 0219-0613
Boney, W.: A Presentation Theorem for Continuous Logic and Metric Abstract Elementary Classes (2014). arXiv:1408.3624
Chang, C.-C., Keisler, H.J.: Continuous Model Theory, Annals of Mathematics Studies, vol. 58. Princeton University Press, Princeton (1966)
Coskey, S., Lupini, M.: A Lopez–Escobar Theorem for Metric Structures, and the Topological Vaught Conjecture (2014). arXiv:1405.2859
Eagle, C.J.: Omitting types for infinitary \([0,1]\)-valued logic. Ann. Pure Appl. Log. 165(3), 913–932 (2014). https://doi.org/10.1016/j.apal.2013.11.006. ISSN: 0168-0072
Henson, C.W.: When do two Banach spaces have isometrically isomorphic nonstandard hulls? Israel J. Math. 22(1), 57–67 (1975). ISSN: 0021-2172
Hirvonen, A.S., Hyttinen, T.:, Metric abstract elementary classes with perturbations. Fund. Math. 217(2), 123–170 (2012). https://doi.org/10.4064/fm217-2-2. https://doi-org.ezp-prod1.hul.harvard.edu/10.4064/fm217-2-2. ISSN: 0016-2736
Iovino, J.: On the maximality of logics with approximations. J. Symb. Log. 66(4), 1909–1918 (2001). https://doi.org/10.2307/2694984. ISSN: 0022-4812
Jech, T.: Set Theory, Springer Monographs in Mathematics, The Third Millennium Edition, Revised and Expanded, p. xiv+769. Springer, Berlin (2003). ISBN: 3-540-44085-2
Keisler, H.J.: Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers, Studies in Logic and the Foundations of Mathematics, Vol. 62, p. x+208. North-Holland Publishing Co., Amsterdam (Loose Erratum) (1971)
Lawvere, F.W.: Metric spaces, generalized logic, and closed categories [Rend. Sem. Mat. Fis. Milano 43(1973), 135–166 (1974); MR0352214 (50 #4701)]. In: Reprints in Theory and Applications of Categories, No. 1, pp. 1–37 (2002). With an author commentary: enriched categories in the logic of geometry and analysis
Shelah, S., Stern, J.: The Hanf number of the first order theory of Banach spaces. Trans. Am. Math. Soc. 244, 147–171 (1978). https://doi.org/10.2307/1997892. https://doi-org.ezp-prod1.hul.harvard.edu/10.2307/1997892. ISSN: 0002-9947
Tent, K., Ziegler, M.:, A Course in Model Theory, Vol. 40, p. x+248. Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9781139015417. https://doi-org.ezp-prod1.hul.harvard.edu/10.1017/CBO9781139015417. ISBN: 978-0-521-76324-0
Acknowledgements
I would like to thank an anonymous referee for detailed and helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ackerman, N.L. Encoding Complete Metric Structures by Classical Structures. Log. Univers. 14, 421–459 (2020). https://doi.org/10.1007/s11787-020-00262-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11787-020-00262-1