Skip to main content Accessibility help




We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic.

For technical reasons we prefer to consider not the valued field (K, |·|) directly, but rather the associated projective spaces KPn, as bounded metric structures.

We show that the class of (projective spaces over) metric valued fields is elementary, with theory MVF, and that the projective spaces Pn and are Pm biinterpretable for every n, m ≥ 1. The theory MVF admits a model completion ACMVF, the theory of algebraically closed metric valued fields (with a nontrivial valuation). This theory is strictly stable (even up to perturbation).

Similarly, we show that the theory of real closed metric valued fields, RCMVF, is the model companion of the theory of formally real metric valued fields, and that it is dependent.



Hide All
[1]Adler, Hans, An introduction to theories without the independence property. Archive for Mathematical Logic, to appear.
[2]Artin, Emil, Algebraic numbers and algebraic functions, Gordon and Breach Science Publishers, New York, 1967.
[3]Yaacov, Itaï Ben, Topometric spaces and perturbations of metric structures. Logic and Analysis, vol. 1 (2008), no. 3–4, pp. 235272, doi:10.1007/s11813-008-0009-x, arXiv:0802.4458.
[4]Yaacov, Itaï Ben, Continuous first order logic for unbounded metric structures. Journal of Mathematical Logic, vol. 8 (2008), no. 2, pp. 197223, doi:10.1142/S0219061308000737, arXiv:0903.4957.
[5]Yaacov, Itaï Ben, On perturbations of continuous structures. Journal of Mathematical Logic, vol. 8 (2008), no. 2, pp. 225249, doi:10.1142/S0219061308000762, arXiv:0802.4388.
[6]Yaacov, Itaï Ben, Continuous and random Vapnik-Chervonenkis classes. Israel Journal of Mathematics, vol. 173 (2009), pp. 309333, doi:10.1007/s11856-009-0094-x, arXiv:0802.0068.
[7]Yaacov, Itaï Ben, Definability of groups in א0-stable metric structures, this Journal, vol. 75 (2010), no. 3, pp. 817–840, doi:10.2178/jsl/1278682202, arXiv:0802.4286.
[8]Yaacov, Itaï Ben and Berenstein, Alexander, On perturbations of Hilbert spaces and probability algebras with a generic automorphism. Journal of Logic and Analysis, vol. 1 (2009), no. 7, pp. 118, doi:10.4115/jla.2009.1.7, arXiv:0810.4086.
[9]Yaacov, Itaï Ben, Berenstein, Alexander, Ward Henson, C., and Usvyatsov, Alexander, Model theory for metric structures, Model theory with applications to algebra and analysis, volume 2 (Chatzidakis, Zoé, Macpherson, Dugald, Pillay, Anand, and Wilkie, Alex, editors), London Math Society Lecture Note Series, vol. 350, Cambridge University Press, 2008, pp. 315427.
[10]Yaacov, Itaï Ben and Usvyatsov, Alexander, Continuous first order logic and local stability. Transactions of the American Mathematical Society, vol. 362 (2010), no. 10, pp. 52135259, doi:10.1090/S0002-9947-10-04837-3, arXiv:0801.4303.
[11]Berkovich, Vladimir, Spectral theory and analytic geometry over non-Archimedean fields. Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990.
[12]Poizat, Bruno, Cours de théorie des modèles, Une introduction á la logique mathèmatique contemporaine, Nur al-Mantiq wal-Ma’rifah, Lyon, 1985.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed