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On the elementary theory of restricted elementary functions

  • Lou van den Dries (a1)
Abstract

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 xRm such that ϕ(x) is true in RRE there is exactly one yRn such that ψ(x, y) is true in RRE. (Here ψ is quantifier free.)

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[D-L] J. Denef and L. Lipshitz , Ultraproducts and approximation in local rings. II, Mathematische Annalen, vol. 253 (1980), pp. 128.

[L-R] L. Lipshitz and L. Rubel , A differentially algebraic replacement theorem, and analog computability, Proceedings of the American Mathematical Society, vol. 99 (1987), pp. 367372.

[vdD 1] L. van den Dries , A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bulletin (New Series) of the American Mathematical Society, vol. 15 (1986), pp. 189193.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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