Definably complete Baire structures

Antongiulio Fornasiero¹ ; Tamara Servi²

¹ Institut für Mathematische Logik Einsteinstr. 62 48149 Münster, Germany
² Centro de Matemática e Aplicações Fundamentais Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal

Fundamenta Mathematicae, Tome 209 (2010) no. 3, pp. 215-241

Résumé

We consider definably complete Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain cannot be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire, and so is every o-minimal expansion of a field. Moreover, unlike the o-minimal case, the structures considered form an axiomatizable class. In this context we prove a version of the Kuratowski–Ulam Theorem, some restricted version of Sard's Lemma and a version of Khovanskii's Finiteness Theorem. We apply these results to prove the o-minimality of every definably complete Baire expansion of an ordered field with any family of definable Pfaffian functions.

Détail

DOI : 10.4064/fm209-3-2

Keywords: consider definably complete baire expansions ordered fields every definable subset domain structure has supremum domain cannot written union definable increasing family nowhere dense sets every expansion real field definably complete baire every o minimal expansion field moreover unlike o minimal structures considered form axiomatizable class context prove version kuratowski ulam theorem restricted version sards lemma version khovanskiis finiteness theorem apply these results prove o minimality every definably complete baire expansion ordered field family definable pfaffian functions