Submission date: 5 August 2019

Abstract:

We give an example of a model of the common theory of o-minimal structures, in a given language, and an expansion of that model admitting the exact same one-variable definable sets with the latter admitting a definable, closed, bounded, and discrete subset and a definable injective self-mapping of that subset which is not surjective. This answers negatively two question by Schoutens; the first being whether there is an axiomatization of the common theory of o-minimal structures in a given language by conditions on one-variable definable sets alone. The second being whether definable completeness and type completeness imply the pigeon-hole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeon-hole principle.

Mathematics Subject Classification: 03C64

Keywords and phrases:

Full text arXiv 1908.01660: pdf,