This volume contains the joint proceedings of two major international logic meetings which took place at the University of Münster, Germany in August 2002: Logic Colloquium '02, the 2002 European Summer Meeting of the Association for Symbolic Logic and Colloquium Logicum 2002, the biannual meeting of the German Association for Mathematical Logic and the Foundations of Exact Sciences. The conferences were attended by international audiences of together more than 300 scientists from five continents, who turned Münster into a logic metropolis for 10 days from August 3 to August 12. The conferences covered all areas of contemporary mathematical logic; philosophical logic and computer science logic were presented as well. The conference programs were densely filled with invited tutorials and plenary talks, several special sessions and a large number of contributed talks. These proceedings contain elaborated and extended versions of a number of invited plenary talks and tutorials. All contributions were subjected to the refereeing standards of the Association of Symbolic Logic.

The town of Münster, located in north-western Germany, is a pleasant middle-sized administrative and university town of 280,000 residents. The city center is characterised by historic buildings, churches, squares and parks, and by an extraordinary number of bicycles on designated cycle paths. In the 1200 year history of Münster, the single most important date was the signing of the Peace of Westphalia in 1648 which marked the end of the Thirty Years War in Europe and can be seen as a very early step towards a united Europe. The University of Münster was first founded in the 18th century. It is now spread out throughout the city and has nearly 40,000 students.

Within the University ofMünster formal andmathematical logic are taught and researched at the Institute for Mathematical Logic and Foundational Research which belongs to the department of mathematics and computer science. The institute was founded and developed by Georg Heinrich Scholz who was appointed to a professorship in philosophy in 1928. The character of formal logic at Münster was shaped by Herrmann Ackermann, Justus Diller, Gisbert Hasenjager, Hans Hermes and Dieter Rodding, with an emphasis on proof theory and recursion theory. Currently the professorships for logic are held by Wolfram Pohlers and Ralf-Dieter Schindler.

Since the day of inception of the Institute for Studies in Theoretical Physics and Mathematics (IPM) in 1989, Mathematical Logic has been one of the main domains of activity at its School of Mathematics.

Briefly, through inviting a select number of prominent logicians from the republics of the former Soviet Union, activities in logic were initiated and the candle of interestwas lit. This startwas relatively successful in kindling interest in the subject in Iran. Furthermore, in those early days, an international congress and a summer school, both in logic, were organized and hosted by IPM, adding to the presence of the subject in Iran. Later, by establishing a Ph.D. program in Mathematical Logic at IPM, the cause was advanced with help from Iranian logicians educated abroad and the program in logic was consolidated. For more details, I refer you to the article in this volume by Professor Larijani, the Director of IPM.

Some three years ago, I proposed to our logic researchers at IPM the possibility of organizing another international workshop. This time our plan was focused on involving the Iranian logicians living abroad and getting help from them. We carried out this plan and it has been a complete success. Support from such logicians was a significant factor in the success of the workshop. I do not intend to add anything further about the workshop since I believe the list of its invited speakers and the list of its presentations provide sufficient manifestation of the success of the workshop.

But I would like to express, with great pleasure, my sincere gratitude to my friends, and IPM's fans, Iraj Kalantari and Ali Enayat. From distant places, they took time to participate in the planning and management of the workshop. Later, they continued in the same way and worked extensively with all, closing the distances through internet and email as if they were virtual officemates at IPM, to arrange for this volume to appear. I also wish to express my thanks to Mojtaba Moniri and Morteza Moniri who led the team of the local organizers.

Introduction. Let V[G] be the result of adding a Cohen real and a Cohen subset of to a model V of the GCH. Every infinite cardinal gets a new subset in V[G], namely, the Cohen real. Yet, in some sense, only and get new subsets. One way to capture this is to say that a sort of dual to Covering holds between V and V[G], namely, “cocovering” for other than and.

We need some definitions.

If is a regular cardinal in V, say that cocovering holds between V and the outer model W if, given any unbounded that lies in W, there exists such that and b is unbounded in.

If V is a standard transitive model of ZFC, say that is an outer model of V if W is also a standard transitive model of ZFC and. In this paper we are only concerned with outer models such that (W;V) satisfies ZFC (in a language with a predicate symbol for V).

Ultimately, ZFC is our metatheory; talk of models can be understood as talk of set, or even countable set models. The reader will note that our results usually can be paraphrased in more traditional terminology, perhaps at the cost of some generality.

This paper proves a sort of converse to the observation with which we began. The general case is not quite so simple as this example suggests. We show that if W is any outer model of V, and sufficient cocovering (and maybe a little covering) hold around, then every subset of that lies in W is set generic over V for a forcing of V-cardinality less than.

Nonstandard analysis is one of the the great achievements of modern applied mathematical logic. In addition to the important philosophical achievement of providing a sound mathematical basis for using infinitesimals in analysis, the methodology is now well established as a tool for both research and teaching, and has become a fruitful field of investigation in its own right. It has been used to discover and prove significant new standard theorems in such diverse areas as probability theory and stochastic analysis, functional analysis, fluid mechanics, dynamical systems and control theory, and recently there have been some striking and unexpected applications to additive number theory.

A conference on Nonstandard Methods and Applications in Mathematics (NS2002) was held in Pisa, Italy from June 12-16 2002. This was originally planned as a special section in the very successful first joint meeting of the American Mathematical Society and the Unione Matematica Italiana. In order to accommodate the large number of mathematicians interested in the field, a satellite conference, hosted by the Università di Pisa and held at the Domus Galilaeana, was added during the days preceding the main AMS/UMI meeting. A complete list of the registered participants appears later in this forward.

This volume is a by product of NS2002. Not a proceedings per se, it is a collection of peer-reviewed papers solicited from some of the participants with the aim of providing something more timely than a textbook, but less ephemeral than a conventional proceedings. To that end, the volume contains both survey papers on topics for which other surveys are either dated or nonexistent, and research articles on applications too recent to have received attention in older volumes.

One of the included papers, on an infinitesimal approach to calculus, deserves special mention. The use of infinitesimals in the teaching of calculus is of course not at all new, though they began to disappear from textbooks late in the 19th century due to concerns about their theoretical underpinnings.