Notre Dame Journal of Formal Logic

Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields

Nicolas Guzy and Cédric Rivière


In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained by Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space.

Article information

Notre Dame J. Formal Logic, Volume 47, Number 3 (2006), 331-341.

First available in Project Euclid: 17 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C10: Quantifier elimination, model completeness and related topics
Secondary: 03C68: Other classical first-order model theory

model companion topological fields geometric axiomatization


Guzy, Nicolas; Rivière, Cédric. Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields. Notre Dame J. Formal Logic 47 (2006), no. 3, 331--341. doi:10.1305/ndjfl/1163775440.

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