Avoiding the projective hierarchy in expansions of the real field by sequences
HTML articles powered by AMS MathViewer
- by Chris Miller PDF
- Proc. Amer. Math. Soc. 134 (2006), 1483-1493 Request permission
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 $\mathbb N$. In particular, let $f\colon \mathbb R\to \mathbb R$ be such that $\lim _{x\to +\infty }f(x)=+\infty$, $f(x)=O(e^{x^N})$ as $x\to +\infty$ for some $N\in \mathbb N$, $(\mathbb R, +,\cdot ,f)$ is o-minimal, and the expansion of $(\mathbb R,+,\cdot )$ by the set $\{ f(k):k\in \mathbb {N} \}$ does not define $\mathbb N$. Then there exist $r>0$ and $P\in \mathbb R[x]$ such that $f(x)=e^{P(x)}(1+O(e^{-rx}))$ as $x\to +\infty$.References
- Michael Boshernitzan, “Orders of infinity” generated by difference equations, Amer. J. Math. 106 (1984), no. 5, 1067–1089. MR 761579, DOI 10.2307/2374273
- 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, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
- Lou van den Dries, o-minimal structures and real analytic geometry, Current developments in mathematics, 1998 (Cambridge, MA), Int. Press, Somerville, MA, 1999, pp. 105–152. MR 1772324
- Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337, DOI 10.1215/S0012-7094-96-08416-1
- Lou van den Dries and Patrick Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4377–4421. MR 1458313, DOI 10.1090/S0002-9947-98-02105-9
- Lou van den Dries and Patrick Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3) 81 (2000), no. 3, 513–565. MR 1781147, DOI 10.1112/S0024611500012648
- Harvey Friedman and Chris Miller, Expansions of o-minimal structures by fast sequences, J. Symbolic Logic 70 (2005), no. 2, 410–418. MR 2140038, DOI 10.2178/jsl/1120224720
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- Chris Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic 68 (1994), no. 1, 79–94. MR 1278550, DOI 10.1016/0168-0072(94)90048-5
- Chris Miller, Exponentiation is hard to avoid, Proc. Amer. Math. Soc. 122 (1994), no. 1, 257–259. MR 1195484, DOI 10.1090/S0002-9939-1994-1195484-5
- 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 and Patrick Speissegger, Pfaffian differential equations over exponential o-minimal structures, J. Symbolic Logic 67 (2002), no. 1, 438–448. MR 1889560, DOI 10.2178/jsl/1190150053
- C. Miller and J. Tyne, Expansions of o-minimal structures by iteration sequences, Notre Dame J. Formal Logic, to appear.
- J.-P. Rolin, P. Speissegger, and A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), no. 4, 751–777. MR 1992825, DOI 10.1090/S0894-0347-03-00427-2
- Maxwell Rosenlicht, Hardy fields, J. Math. Anal. Appl. 93 (1983), no. 2, 297–311. MR 700146, DOI 10.1016/0022-247X(83)90175-0
- Maxwell Rosenlicht, The rank of a Hardy field, Trans. Amer. Math. Soc. 280 (1983), no. 2, 659–671. MR 716843, DOI 10.1090/S0002-9947-1983-0716843-5
- Maxwell Rosenlicht, Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc. 299 (1987), no. 1, 261–272. MR 869411, DOI 10.1090/S0002-9947-1987-0869411-2
Additional Information
- Chris Miller
- Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
- MR Author ID: 330760
- Email: miller@math.ohio-state.edu
- 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.
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2199196