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PSEUDO-FINITE SETS, PSEUDO-O-MINIMALITY

Part of: Graph theory Extremal combinatorics Model theory

Published online by Cambridge University Press:  26 October 2020

NADAV MEIR Open the ORCID record for NADAV MEIR [Opens in a new window]
NADAV MEIR*
Affiliation:
DEPARTMENT OF MATHEMATICS BEN GURION UNIVERSITY OF THE NEGEV P.O.B. 653, BE’ER SHEVA 8410501, ISRAEL and INSTYTUT MATEMATYCZNY, UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4, 50-384 WROCŁAW, POLAND E-mail: mein@math.bgu.ac.il
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Abstract

We give an example of two ordered structures $\mathcal {M},\mathcal {N}$ in the same language $\mathcal {L}$ with the same universe, the same order and admitting the same one-variable definable subsets such that $\mathcal {M}$ is a model of the common theory of o-minimal $\mathcal {L}$-structures and $\mathcal {N}$ admits 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 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.

Keywords

definably completetype completepseudo-o-minimalo-minimalismpigeonhole principle

MSC classification

Primary: 03C10: Quantifier elimination, model completeness and related topics
Secondary: 03C35: Categoricity and completeness of theories 05D10: Ramsey theory 05C55: Generalized Ramsey theory

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 2 , June 2021 , pp. 577 - 599
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Copyright
© The Association for Symbolic Logic 2020

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Footnotes

*

The second affiliation for the author of the article has been corrected. An erratum detailing this change has also been published (doi: 10.1017/jsl.2021.100).

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