**Pseudo-o-minimality and pseudo-finite sets.**

Given a language L, the class of o-minimal L-structures is not elementary, e.g., an ultraproduct of o-minimal structures need not be o-minimal. This gives rise to the following notion introduced by Hans Schoutens: Given a language L, an L-structure is pseudo-o-minimal if it satisfies the common theory of o-minimal L-structures. Of particular importance in pseudo-o-minimal structures are pseudo-finite sets. A definable set in an ordered structure is pseudo-finite if it is closed, bounded and discrete. In this talk, we will review research on first-order properties of pseudo-finite structures, such as the intermediate value property and local o-minimality. (To be defined in the talk.) Finally, we will see how pseudo-finite sets can be used to answer two questions by Schoutens, one of them asking whether there is an axiomatization of pseudo-o-minimality by first-order conditions on one-variable formulae only. This will also partially answer a conjecture by Antongiulio Fornasiero.

No background in o-minimality will be assumed.

## Date:

Wed, 29/05/2019 - 11:00 to 13:00

## Location:

Ross 63