Defining additive subgroups of the reals from convex subsets
HTML articles powered by AMS MathViewer
- by Michael A. Tychonievich PDF
- Proc. Amer. Math. Soc. 137 (2009), 3473-3476 Request permission
Abstract:
Let $G$ be a subgroup of the additive group of real numbers and let $C\subseteq G$ be infinite and convex in $G$. We show that $G$ is definable in $(\mathbb R,+,\cdot ,C)$ and that ${\mathbb Z}$ is definable if $G$ has finite rank. This has a number of consequences for expansions of certain o-minimal structures on the real field by multiplicative groups of complex numbers.References
- Ricardo Bianconi, Nondefinability results for expansions of the field of real numbers by the exponential function and by the restricted sine function, J. Symbolic Logic 62 (1997), no. 4, 1173–1178. MR 1617985, DOI 10.2307/2275634
- Lou van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 189–193. MR 854552, DOI 10.1090/S0273-0979-1986-15468-6
- Lou van den Dries, Dense pairs of o-minimal structures, Fund. Math. 157 (1998), no. 1, 61–78. MR 1623615, DOI 10.4064/fm-157-1-61-78
- Lou van den Dries and Ayhan Günaydın, The fields of real and complex numbers with a small multiplicative group, Proc. London Math. Soc. (3) 93 (2006), no. 1, 43–81. MR 2235481, DOI 10.1017/S0024611506015747
- Ayhan Günaydın, Model theory of fields with multiplicative groups, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2008.
- Chris Miller, Tameness in expansions of the real field, Logic Colloquium ’01, Lect. Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 281–316. MR 2143901
- Chris Miller, Avoiding the projective hierarchy in expansions of the real field by sequences, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1483–1493. MR 2199196, DOI 10.1090/S0002-9939-05-08112-8
- Chris Miller and Patrick Speissegger, A trichotomy for expansions of $\mathbb R_{\textrm {an}}$ by trajectories of analytic planar vector fields, preliminary report, available at http://www.math.ohio-state.edu/~miller.
- Bjorn Poonen, Uniform first-order definitions in finitely generated fields, Duke Math. J. 138 (2007), no. 1, 1–22. MR 2309154, DOI 10.1215/S0012-7094-07-13811-0
- Julia Robinson, The undecidability of algebraic rings and fields, Proc. Amer. Math. Soc. 10 (1959), 950–957. MR 112842, DOI 10.1090/S0002-9939-1959-0112842-7
- Raphael M. Robinson, The undecidability of pure transcendental extensions of real fields, Z. Math. Logik Grundlagen Math. 10 (1964), 275–282. MR 172803, DOI 10.1002/malq.19640101803
Additional Information
- Michael A. Tychonievich
- Affiliation: Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
- Email: tycho@math.ohio-state.edu
- Received by editor(s): October 1, 2008
- Received by editor(s) in revised form: December 22, 2008, and February 14, 2009
- Published electronically: May 8, 2009
- Communicated by: Julia Knight
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3473-3476
- MSC (2000): Primary 03C64; Secondary 14P10
- DOI: https://doi.org/10.1090/S0002-9939-09-09914-6
- MathSciNet review: 2515416