Defining additive subgroups of the reals from convex subsets

Author:
Michael A. Tychonievich

Journal:
Proc. Amer. Math. Soc. **137** (2009), 3473-3476

MSC (2000):
Primary 03C64; Secondary 14P10

Published electronically:
May 8, 2009

MathSciNet review:
2515416

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a subgroup of the additive group of real numbers and let be infinite and convex in . We show that is definable in and that is definable if 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.

**[1]**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**, 10.2307/2275634**[2]**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**, 10.1090/S0273-0979-1986-15468-6**[3]**Lou van den Dries,*Dense pairs of o-minimal structures*, Fund. Math.**157**(1998), no. 1, 61–78. MR**1623615****[4]**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**, 10.1017/S0024611506015747**[5]**Ayhan Günaydın,*Model theory of fields with multiplicative groups*, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2008.**[6]**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****[7]**Chris Miller,*Avoiding the projective hierarchy in expansions of the real field by sequences*, Proc. Amer. Math. Soc.**134**(2006), no. 5, 1483–1493 (electronic). MR**2199196**, 10.1090/S0002-9939-05-08112-8**[8]**Chris Miller and Patrick Speissegger,*A trichotomy for expansions of by trajectories of analytic planar vector fields*, preliminary report, available at`http://www.math.ohio-state.edu/~miller`.**[9]**Bjorn Poonen,*Uniform first-order definitions in finitely generated fields*, Duke Math. J.**138**(2007), no. 1, 1–22. MR**2309154**, 10.1215/S0012-7094-07-13811-0**[10]**Julia Robinson,*The undecidability of algebraic rings and fields*, Proc. Amer. Math. Soc.**10**(1959), 950–957. MR**0112842**, 10.1090/S0002-9939-1959-0112842-7**[11]**Raphael M. Robinson,*The undecidability of pure transcendental extensions of real fields*, Z. Math. Logik Grundlagen Math.**10**(1964), 275–282. MR**0172803**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
03C64,
14P10

Retrieve articles in all journals with MSC (2000): 03C64, 14P10

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

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-09914-6

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

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.