Premium
Definable types in algebraically closed valued fields
Author(s) -
Cubides Kovacsics Pablo,
Delon Françoise
Publication year - 2016
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201400039
Subject(s) - algebraically closed field , counterexample , mathematics , type (biology) , property (philosophy) , statement (logic) , field (mathematics) , pure mathematics , discrete mathematics , ecology , philosophy , epistemology , political science , law , biology
In [15][D. Marker, 1994], Marker and Steinhorn characterized models M ≺ N of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models M ≺ N for which all 1‐types over M realized in N are definable has already the desired property. Although it is true that if M is an algebraically closed valued field such that all 1‐types over M are definable then all types over M are definable, we build a counterexample for the relative statement, i.e., we show for any n ≥ 1 that there is a pair M ≺ N of algebraically closed valued fields such that all n ‐types over M realized in N are definable but there is an n + 1 ‐type over M realized in N which is not definable.