We continue work of Shelah on the cardinality of families of pairwise incompatible types in simple theories obtaining characterizations of simple and supersimple theories. We develop a local analysis of the number of types in simple theories and we find a new example of a simple unstable theory.

I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski-Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.

I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski-Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.

A Steiner triple system (STS) is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite STS has a Fraïssé limit M_F. Here, we show that the theory T of M_F is the model completion of the theory of STSs. We also prove that T is not small and it has quantifier elimination, (...) TP2, NSOP1, elimination of hyperimaginaries and weak elimination of imaginaries. (shrink)

We give an exposition of the compactness of L(QcfC), for any set C of regular cardinals.

Our aim is to solve a quite old question on the difference between expandability and compact expandability. Toward this, we further investigate the logic of countable cofinality.

We characterize omissibility of a type, or a family of types, in a countable theory in terms of non-existence of a certain tree of formulas. We extend results of L. Newelski on omitting $ non-isolated types. As a consequence we prove that omissibility of a family of $ types is equivalent to omissibility of each countable subfamily.

We study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A ⊆ B ⊆ acl and closed subgroups of the Galois group Aut/A). We also characterize when the topology of the Galois group is the quotient topology.

This paper presents an example of a supersimple nonlow theory and characterizes its independence relation.

We study local strengthenings of the simplicity condition. In particular, we define and study a local Lascar rank, as well as short, low, supershort and superlow theories. An example of a low, non supershort theory is given.

We introduce the notion of normal hyperimaginary and we develop its basic theory. We present a new proof of the Lascar-Pillay theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries. However, the use of the Peter–Weyl theorem on the structure of compact Hausdorff groups is not completely eliminated from the proof. In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of φ-types for bounded (...) hyperimaginaries. (shrink)

A model $M$ of countable similarity type and cardinality $\kappa$ is expandable if every consistent extension $T_{1}$ of its complete theory with $|T_{1}|\leq \kappa$ is satisfiable in $M$ and it is compactly expandable if every such extension which additionally is finitely satisfiable in $M$ is satisfiable in $M$ . In the countable case and in the case of a model of cardinality $\geq 2^{\omega}$ of a superstable theory without the finite cover property the notions of saturation, expandability and compactness for (...) expandability agree. The question of the existence of compactly expandable models which are not expandable is open. Here we present a test which serves to prove that a compactly expandable model of cardinality $\geq 2^{\omega}$ of a superstable theory is expandable. It is stated in terms of the existence of a certain elementary submodel whose corresponding theory of pairs of models satisfies a weak elimination of Ramsey quantifiers. (shrink)

We obtain a chain condition for dividing in an arbitrary theory and a new and shorter proof of a chain condition result of Shelah for simple theories.

Abstract.We obtain a chain condition for dividing in an arbitrary theory and a new and shorter proof of a chain condition result of Shelah for simple theories.

We prove that a theory T has stable forking if and only if Teq has stable forking.

We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs.

In this paper we show that in every |T |+-resplendent model N , for every A ⊆ N such that |A | ≤ |T |, the group Autf of strong automorphisms is the least very normal subgroup of the group Aut and the quotient Aut/Autf is the Lascar group over A . Then we generalize this result to every |T |+-saturated and strongly |T |+-homogeneous model.