Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-15T00:59:51.272Z Has data issue: false hasContentIssue false
  • English
  • Français

On the elementary theory of restricted elementary functions

Published online by Cambridge University Press:  12 March 2014

Lou van den Dries
Lou van den Dries*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Get accessRights & Permissions [Opens in a new window]

Extract

As a contribution to definability theory in the spirit of Tarski's classical work on (R, <, 0, 1, +, ·) we extend here part of his results to the structure

Here exp ∣[0, 1] and sin ∣[0, π] are the restrictions of the exponential and sine function to the closed intervals indicated; formally we identify these restricted functions with their graphs and regard these as binary relations on R. The superscript “RE” stands for “restricted elementary” since, given any elementary function, one can in general only define certain restrictions of it in RRE.

Let (RRE, constants) be the expansion of RRE obtained by adding a name for each real number to the language. We can now formulate our main result as follows.

Theorem. (RRE, constants) is strongly model-complete.

This means that every formula ϕ(X1, …, Xm) in the natural language of (RRE, constants) is equivalent to an existential formula

with the extra property that for each x ∈ Rm such that ϕ(x) is true in RRE there is exactly one y ∈ Rn such that ψ(x, y) is true in RRE. (Here ψ is quantifier free.)

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988, pp. 796 - 808
Copyright
Copyright © Association for Symbolic Logic 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[D-L]Denef, J. and Lipshitz, L., Ultraproducts and approximation in local rings. II, Mathematische Annalen, vol. 253 (1980), pp. 1–28.CrossRefGoogle Scholar
[D-vdD]Denef, J. and van den Dries, L., P-adic and real subanalytic sets, Annals of Mathematics (to appear).Google Scholar
[G-R]Gunning, R. and Rossi, H., Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.Google Scholar
[K]Kaplansky, I., An introduction to differential algebra, Hermann, Paris, 1957.Google Scholar
[L-R]Lipshitz, L. and Rubel, L., A differentially algebraic replacement theorem, and analog computability, Proceedings of the American Mathematical Society, vol. 99 (1987), pp. 367–372.CrossRefGoogle Scholar
[S]Seidenberg, A., An elimination theory for differential algebra, University of California Publications in Mathematics, New Series, vol. 3 (1956), pp. 31–66.Google Scholar
[vdD 1]van den Dries, L., A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bulletin (New Series) of the American Mathematical Society, vol. 15 (1986), pp. 189–193.CrossRefGoogle Scholar
[vdD2]van den Dries, L., The logic of Weierstrass preparation (to appear).Google Scholar
[Z-S]Zariski, O. and Samuel, P. (with Cohen, I. S.), Commutative algebra, Vol. II, 2nd ed., Springer-Verlag, New York, 1975.Google Scholar