Externally definable sets and dependent pairs

On externally definable sets and a theorem of Shelah

In [4] Saharon Shelah proves the striking and important result that if T is a complete theory without the independence property,M is a model of T , and M∗ is the expansion of M by adjoining predicates for “externally definable” subsets of M,M ×M, ..., then Th(M∗) has quantifier elimination. Shelah’s proof is not so long, but arguably rather obscure. The aim of this expository paper is to give a...

Ind- and pro- definable sets

We describe the indand procategories of the category of definable sets, in some first order theory, in terms of points in a sufficiently saturated model.

Definable Sets in Ordered Structures

This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ^-minimal structures. The definition of this class and the corresponding class of theories, the strongly ©-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizati...

Definable Sets in Valued Fields

On Pairs of Definable Orthogonal Families

We introduce the notion of an M-family of infinite subsets of N which is implicitly contained in the work of A. R. D. Mathias. We study the structure of a pair of orthogonal hereditary families A and B, where A is analytic and B is C-measurable and an M-family.