Pseudofinite sets, pseudoominimality
Abstract
We give an example of two ordered structures M, N in the same language L with the same universe, the same order and admitting the same onevariable definable subsets such that M is a model of the common theory of ominimal Lstructures and N admits a definable, closed, bounded, and discrete subset and a definable injective selfmapping of that subset which is not surjective. This answers negatively two questions by Schoutens; the first being whether there is an axiomatization of the common theory of ominimal structures in a given language by conditions on onevariable definable sets alone. The second being whether definable completeness and type completeness imply the pigeonhole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeonhole principle.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 DOI:
 10.48550/arXiv.1908.01660
 arXiv:
 arXiv:1908.01660
 Bibcode:
 2019arXiv190801660M
 Keywords:

 Mathematics  Logic;
 03C64
 EPrint:
 21 pages