Abstract not found

We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conjecture over a number field. We conclude with some other applications.

Abstract. This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (“see... ” means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall ’97 and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Ros̷lanowski for many helpful comments. (666) revision:2001-11-12 modified:2003-11-18

Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new.

Our theme is that not every interesting question in set theory is independent of ZF C. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 without CH; we prove in ZF C a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove — again in ZFC — that for a large class of cardinals there is no universal linear order (e.g. in every ℵ1 < λ < 2 ℵ0). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles ” ℵ1 — a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the non-existence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

Finite-model theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finite-model theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:

We consider relational databases organized over an ordered domain with some additional relations---a typical example is the ordered domain of rational numbers together with the operation of addition. In the focus of our study are the first-order (FO) queries that are invariant under order-preserving "permutations"---such queries are called ordergeneric. It has recently been discovered that for some domains ordergeneric FO queries fail to express more than pure order queries. For example, every order-generic FO query over rational numbers with + can be rewritten without +. For some other domains, however, this is not the case. We provide very general conditions on the FO theory of the domain that ensure the collapse of order-generic extended FO queries to pure order queries over this domain: the Pseudo-finite Homogeneity Property and a stronger Isolation Property. We further distinguish one broad class of domains satisfying the Isolation Property, the so-called quasi-o -...

. We study model theoretical stability for Banach spaces and structures based on Banach spaces, e.g., Banach lattices or C # -algebras. We prove that a theory is stable if and only if the following condition is true in every model E of the theory: If ( am ) and ( b n ) are bounded sequences in E k and E l (respectively) and R : E k E l # R is definable, then there exist subsequences ( a m i ) and ( b n ) such that lim i ## lim j ## R( a m i , b n j ) = lim j ## lim i ## R( a m , b n ). 1. INTRODUCTION A framework for the model theoretical analysis of various structures from functional analysis was introduced in the monograph [6]. The class of structures under consideration includes Banach spaces as well as further structures from functional analysis which are based on Banach spaces e.g., C # -algebras. The model theoretical language of [6] provides tools to prove new results for important classes of structures from analysis. In this...

Abstract. We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CM-trivial. 1.

Abstract not found