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On definability of types and relative stability
Author(s) -
Verbovskiy Viktor
Publication year - 2019
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.201600084
Subject(s) - mathematics , type (biology) , stability (learning theory) , model theory , order (exchange) , type theory , pure mathematics , discrete mathematics , computer science , machine learning , ecology , finance , economics , biology
Abstract In this paper, we consider the question of definability of types in non‐stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to Δ if any Δ‐type over a model has few extensions up to complete types. We prove that an n ‐type over a model of a theory that is stable up to Δ is definable if and only if its Δ‐part is definable.