Formalization has three roles: 1) a foundation for an area (perhaps all) of mathematics, 2) a resource for investigating problems in "normal" mathematics, 3) a tool to organize various mathematical areas so as to emphasize commonalities and differences. We focus on the use of theories and syntactical properties of theories in roles 2) and 3). We regard a property of a theory (in first or second order logic) as virtuous if the property has mathematical consequences for the theory or for models of the theory. We rehearse some results of Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, "categoricity" has little virtue. For first order logic, categoricity is trivial. But "categoricity in power" illustrates the sort of mathematical consequences we mean. One can lay out a schema with a few parameters (depending on the theory) which describes the structure of any model of any theory categorical in uncountable power. Similar schema for the decomposition of models apply to other theories according to properties defining the stability hierarchy. We describe arguments using properties, which essentially involve formalizing mathematics, to obtain results in "mainsteam" mathematics. We consider discussions on method by Kazhdan, Bourbaki, and Weil as well as such logicians as Macintyre and Shelah. Paper

Given a group, one may wonder what are the fine properties of its most generic elements. This question remains open in most contexts usually considered in model theory, from stable to NIP theories. I will review what is known about the question for groups of finite Morley rank, stable free product of groups, groups definable over o-minimal structures or p-adic fields.

In the search for new axioms there has been a renewed interest in reflection principles, in part because they provide the best candidates -- indeed some (like Tait) would say the only candidates -- for axioms that are intrinsically justified on the basis of our conception of set. I will discuss a series of limitative results concerning such principles, results that collectively show that general reflection principles (in the sense of Levy, Bernays, Goedel, Tait, and others) are either weak (in that they are consistent relative to the Erdos cardinal \kappa(\omega)) or inconsistent. I will then discuss other work concerning principles that are often called "reflection principles" -- such as older work of Reinhardt and Magidor and more recent work of Welch and Bagaria -- and argue that they belong to a fundamentally different class of principles and rest on different conceptions, conceptions that are best labeled "extension principles" or "resemblance principles". Here, in contrast to the case of reflection principles, the case that they are intrinsically justified on the basis of our conception of set is even more doubtful. I take these conclusions to underscore the importance of extrinsic justifications.

A generalized logic is a mechanism of extending first order logic in order to express properties of mathematical structures or of elements of such structures. A prime example is higher order logic, like second order logic. A generalized logic is typically sensitive to the set theoretical frame work in which the mathematical structure is embedded. This creates an interesting interplay between properties of the logic, like compactness, Skolem-Lowenhiem theorems etc. and the underlying Set Theory.

The interaction can go both ways: A desired properties of the logic under consideration can be used as a motivation for new axioms for Set Theory, for new notions of large cardinals, etc. On the other hand analysis of the possible properties of a given logic in different set theoretic universes can give some insight into the the strength of the logic and its relations with other logics. For instance a typical (admittedly vague) problem is the extent by which a given logic is really logic, or is it Set Theory in disguise.

In this talk we shall show some examples of such interplay. The talk will assume only very basic acquaintance with Model Theory and Set Theory, so hopefully it should accessible to any logic graduate student.

Flag algebras is a recently developed method for approaching, in a natural way, problems of a certain kind in extremal combinatorics. The backbone of this method is made by special commutative algebras endowed with a rich additional structure. And a significant part of this stricture either directly uses the language of mathematical logic or bears an intriguing resemblance to the model theory.

In this talk we will discuss the method itself and its potential connections to logic (the latter part will probably be a bit speculative). If time permits, we will also review some concrete combinatorial results obtained with the help of flag algebras.