# Pseudo-finite sets, pseudo-o-minimality

## TL;DR

The paper constructs two ordered structures with identical one-variable definable sets, illustrating limitations in axiomatizing o-minimal theories and implications for definable completeness and the pigeonhole principle.

## Contribution

It provides a counterexample showing that definable completeness does not imply the pigeonhole principle, challenging previous assumptions in o-minimal theory.

## Key findings

- Counterexample to axiomatization of o-minimal theories with one-variable sets
- Definable completeness does not imply the pigeonhole principle
- Partial answer to questions by Schoutens and Fornasiero

## 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 one-variable definable subsets such that M is a model of the common theory of o-minimal L-structures and 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 questions 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.