Following up on a talk by Roman Kossak earlier this semester, I will discuss work by Engstrom and Kaye which address the question of existence of transplendent models (models with expansions omitting a type). If there is time, I will talk about transplendent models of PA.
I will review the concept of vc-dimension of a formula, and the vc-function of a first order theory. The concept of breadth on the lattice of PP-definable subgroups of a module will be defined, and the relationship between these notions will be explored. Some model theory of Modules will be used to refine certain questions from the paper of Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko. The talk will include several pictures and examples to clarify the notions involved.
For a fixed language L, the first-order L-theory of o-minimality is the set of those L-sentences true in all o-minimal L-structures. It follows from a classical model-theoretic result that a model of this theory is either o-minimal or an elementary substructure of an ultraproduct of o-minimal L-structures. We call these structures pseudo-o-minimal, and note that they will generally not be o-minimal.
In this talk, I will discuss how the study of pseudo-o-minimality fits in to the ongoing project of classifying the tame weakenings of o-minimality. My main focus will be on the recent question of whether for certain fixed languages L, the first-order L-theory of o-minimality is recursively axiomatizable. I will show that it is not whenever L extends the language of ordered fields by at least one new predicate or function symbol. With the time remaining, I will outline some of what is known about the relative tameness of pseudo-o-minimal structures, and mention some open problems in the area.
In the early 1980s, after Khovanskii’s ICM lecture, van den Dries formulated the conjecture that the expansion P of the real field by all pfaffian functions was model complete. Thinking about the problem led him to formulate a minimality notion in expansions of the real order, which directly inspired Pillay and Steinhorn in their discovery of o-minimality. However, while P has been known to be o-minimal since Wilkie’s groundbreaking work in 1996, van den Dries’s conjecture is still open today. Recently, Lion and I proved a variant of this conjecture, in which “pfaffian functions” are replaced with “nested Rolle leaves”, which in essence correspond to the objects originally studied by Khovanskii. The mystery lies in how these two expansions are related. I will explain each of them and exhibit a third related notion, found recently in joint work with Jones, which might clarify this relationship.
I will discuss an observation Ivo Herzog and I made in the last millennium that yields a purely topological definition of stability of a complete first-order theory in terms of their Stone spaces.
I will explain why countable models of PA which are just recursively saturated do not have maximal automorphisms. If time permits I will also show why recursive saturation implies standard system saturation for models of rich theories.
A maximal automorphism of a structure M is an automorphism under which no non-algebraic element of M is fixed. A problem which has attracted some attention is when for a theory T any countable recursively saturated model of T has a maximal automorphism. In this talk I will review what is known about this problem in various contexts and then prove a general result that guarantees, under certain mild conditions on T, that any countable recursively saturated model of T does indeed have a maximal automorphism.
Resplendence is a very useful form of second order saturation. Transplendence, introduced by Fredrik Engström and Richard Kaye, is a stronger notion, that guarantees existence of expansions omitting a type. I will give motivation and outline Engström and Kaye’s general theory of transplendent structures.
We consider maximal independent sets within various sorts of groups and fields freely generated by countably many generators. The simplest example is the free divisible abelian group, which is just an infinite-dimensional rational vector space. As one moves up to free abelian groups, free groups, and “free fields” (i.e. purely transcendental field extensions), maximal independent sets and independent generating sets both become more complicated, from the point of view of computable model theory, but sometimes in unpredictable ways, and certain questions remain open. We present the topic partly for its own sake, but also with the intention of introducing the techniques of computable model theory and illustrating some of its possible uses for an audience to which it may be unfamiliar.
This is joint work with Charles McCoy.
We’ll meet to discuss topics to be covered for the semester.