Omitting types in logic of metric structures

2014/11/11 by Farah, Ilijas, Magidor, Menachem
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1411.2987

Abstract

This paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenstein, Henson and Usvyatsov. While a complete type is omissible in some model of a countable complete theory if and only if it is not principal, this is not true for the incomplete types by a result of Ben Yaacov. We prove that there is no simple test for determining whether a type is omissible in a model of a theory T in a countable language. More precisely, we find a theory in a countable language such that the set of types omissible in some of its models is a complete Σ¹₂ set and a complete theory in a countable language such that the set of types omissible in some of its models is a complete Π¹₁ set. Two more unexpected examples are given: (i) a complete theory T and a countable set of types such that each of its finite sets is jointly omissible in a model of T, but the whole set is not and (ii) a complete theory and two types that are separately omissible, but not jointly omissible, in its models.

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