Interpreting the projective hierarchy in expansions of the real line
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- by Philipp Hieronymi and Michael Tychonievich PDF
- Proc. Amer. Math. Soc. 142 (2014), 3259-3267 Request permission
Abstract:
We give a criterion when an expansion of the ordered set of real numbers defines the image of $(\mathbb {R},+,\cdot ,\mathbb {N})$ under a semialgebraic injection. In particular, we show that for a non-quadratic irrational number $\alpha$, the expansion of the ordered $\mathbb {Q}(\alpha )$-vector space of real numbers by $\mathbb {N}$ defines multiplication on $\mathbb {R}$.References
- Tom M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics, vol. 41, Springer-Verlag, New York, 1990. MR 1027834, DOI 10.1007/978-1-4612-0999-7
- Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. MR 0422171, DOI 10.1017/CBO9780511565977
- Harvey Friedman and Chris Miller, Expansions of o-minimal structures by sparse sets, Fund. Math. 167 (2001), no. 1, 55–64. MR 1816817, DOI 10.4064/fm167-1-4
- Philipp Hieronymi, Defining the set of integers in expansions of the real field by a closed discrete set, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2163–2168. MR 2596055, DOI 10.1090/S0002-9939-10-10268-8
- Chris Miller, Expansions of dense linear orders with the intermediate value property, J. Symbolic Logic 66 (2001), no. 4, 1783–1790. MR 1877021, DOI 10.2307/2694974
- Carlo Toffalori and Kathryn Vozoris, Notes on local o-minimality, MLQ Math. Log. Q. 55 (2009), no. 6, 617–632. MR 2582162, DOI 10.1002/malq.200810016
- Volker Weispfenning, Mixed real-integer linear quantifier elimination, Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC), ACM, New York, 1999, pp. 129–136. MR 1802076, DOI 10.1145/309831.309888
Additional Information
- Philipp Hieronymi
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
- MR Author ID: 894309
- Email: P@hieronymi.de
- Michael Tychonievich
- Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
- Email: tycho@math.ohio-state.edu
- Received by editor(s): August 1, 2012
- Received by editor(s) in revised form: September 13, 2012
- Published electronically: March 19, 2014
- Communicated by: Julia Knight
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 3259-3267
- MSC (2010): Primary 03C64
- DOI: https://doi.org/10.1090/S0002-9939-2014-12023-5
- MathSciNet review: 3223381