Avoiding the projective hierarchy in expansions of the real field by sequences

Author:
Chris Miller

Journal:
Proc. Amer. Math. Soc. **134** (2006), 1483-1493

MSC (2000):
Primary 03C64; Secondary 26A12

DOI:
https://doi.org/10.1090/S0002-9939-05-08112-8

Published electronically:
October 5, 2005

MathSciNet review:
2199196

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define . In particular, let be such that , as for some , is o-minimal, and the expansion of by the set does not define . Then there exist and such that as .

**[1]**M. Boshernitzan,*Discrete ``orders of infinity''*, Amer. J. Math.**106**(1984), no. 5, 1147-1198.MR**0761583 (86f:12002b)****[2]**L. 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**0854552 (88b:03048)****[3]**-,*Tame topology and o- minimal structures*, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.MR**1633348 (99j:03001)****[4]**-,*o- minimal structures and real analytic geometry*, Current developments in mathematics, 1998 (Cambridge, MA), 1999, pp. 105-152.MR**1772324 (2001j:03075)****[5]**L. van den Dries and C. Miller,*Geometric categories and o-minimal structures*, Duke Math. J.**84**(1996), no. 2, 497-540.MR**1404337 (97i:32008)****[6]**L. van den Dries and P. Speissegger,*The real field with convergent generalized power series*, Trans. Amer. Math. Soc.**350**(1998), no. 11, 4377-4421.MR**1458313 (99a:03036)****[7]**-,*The field of reals with multisummable series and the exponential function*, Proc. London Math. Soc. (3)**81**(2000), no. 3, 513-565.MR**1781147 (2002k:03057)****[8]**H. Friedman and C. Miller,*Expansions of o- minimal structures by fast sequences*, J. Symbolic Logic**70**(2005), no. 2, 410-418. MR**2140038****[9]**A. Kechris,*Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.MR**1321597 (96e:03057)****[10]**C. Miller,*Expansions of the real field with power functions*, Ann. Pure Appl. Logic**68**(1994), no. 1, 79-94.MR**1278550 (95i:03081)****[11]**-,*Exponentiation is hard to avoid*, Proc. Amer. Math. Soc.**122**(1994), no. 1, 257-259.MR**1195484 (94k:03042)****[12]**-,*Tameness in expansions of the real field*, Logic Colloquium '01 (Vienna, 2001), Lecture Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, pp. 281-316. MR**2143901****[13]**C. Miller and P. Speissegger,*Pfaffian differential equations over exponential o- minimal structures*, J. Symbolic Logic**67**(2002), no. 1, 438-448.MR**1889560 (2003a:03056)****[14]**C. Miller and J. Tyne,*Expansions of o-minimal structures by iteration sequences*, Notre Dame J. Formal Logic, to appear.**[15]**J.-P. Rolin, P. Speissegger, and A. Wilkie,*Quasianalytic Denjoy-Carleman classes and o- minimality*, J. Amer. Math. Soc.**16**(2003), no. 4, 751-777.MR**1992825 (2004g:14065)****[16]**M. Rosenlicht,*Hardy fields*, J. Math. Anal. Appl.**93**(1983), no. 2, 297-311.MR**0700146 (85d:12001)****[17]**-,*The rank of a Hardy field*, Trans. Amer. Math. Soc.**280**(1983), no. 2, 659-671.MR**0716843 (85d:12002)****[18]**-,*Growth properties of functions in Hardy fields*, Trans. Amer. Math. Soc.**299**(1987), no. 1, 261-272.MR**0869411 (88b:12010)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
03C64,
26A12

Retrieve articles in all journals with MSC (2000): 03C64, 26A12

Additional Information

**Chris Miller**

Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210

Email:
miller@math.ohio-state.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-08112-8

Received by editor(s):
February 4, 2004

Received by editor(s) in revised form:
November 17, 2004

Published electronically:
October 5, 2005

Additional Notes:
This research was partially supported by NSF Grant No. DMS-9988855.

Communicated by:
Carl G. Jockusch, Jr.

Article copyright:
© Copyright 2005
American Mathematical Society

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