# Program for Friday

W. Hugh Woodin, University of California, Berkeley.

** Hour: ** 9:00 - 10:00

** Title: ** The Mostowski Collapse and the Inner Model Program.

** Abstract: ** The Mostowski collapse is ubiquitous in modern Set Theory. One important manifestation is in condensation principles which lie at the core of the Inner Model Program. These condensation principles are really generalizations of the Mostowski collapse and the basic open problems of the Inner Model Program can be reformulated in terms of such principles.

Thomas Jech, Pennsylvania State University and Czech Academy of Sciences.

** Hour: ** 10:20 - 11:20

** Title: ** Measure algebras.

** Abstract: ** A measure algebra is a complete Boolean algebra carrying
a countably additive measure. We give a description of
measure algebras in Boolean algebraic terms. The work
was inspired by a problem of John von Neumann that he
stated in 1937 in the Scottish book.

Menachem Magidor, Hebrew University.

** Hour: ** 11:30 - 12:30

** Title: ** Inner Models constructed from generalized logics.

** Abstract: ** This talk combines two subjects which were very much in the centre of Mostowski's interests: Generalized logics and quantifiers and Set Theory. In a joint work with J.Kennedy and J. Väänänen we study the inner models of Set Theory that are obtained like the constructible universe where each stage of the construction is obtained from the previous stage by taking all definable subsets of the last stage. We get a rich collection of inner models by changing the notion of “definable” from “first order definable” to “definable by some generalized logic”. The study of these inner models yields some interesting inner models and it leads to some intriguing problems.

Angus Macintyre, Queen Mary College, University of London.

** Hour: ** 14:30 - 15:30

** Title: ** The Elementary Theory of the Adele Ring over a Number Field (joint work with Jamshid Derakhshan).

** Abstract: ** We greatly refine work of Weisspfenning on the adeles by internalizing to ring theory the Feferman-Vaught work on restricted products( inspired in turn by Mostowski's work on products). In particular we get very precise quantifier- eliminations, using advanced work on the model theory of Henselian fields. We prove measurability of definable sets, but show that definable is not the same as locally closed. We present some extensions both of the Feferman-Vaught work,and of the Weisspfenning work, inspired by quadratic reciprocity and more general reciprocity theorems. We pay attention to uniformity over all number fields.

Mikołaj Bojańczyk, University of Warsaw.

** Hour: ** 15:50 - 16:50

** Title: ** Computation in Sets With Atoms.

** Abstract: ** The abstract of this talk is avaliable as a separate .pdf file.

Harvey Friedman, Ohio State University.

** Hour: ** 17:00 - 18:00

** Title: ** Concrete mathematical incompleteness.

** Abstract: ** Mathematicians view mathematics as a special subject with singular attractive features. Most feel that the great power and
stability of the “rule book for mathematics” is an important component of their relationship with mathematics. They believe that
the rules are so powerful and stable that they do not have to remember or refer to it.

The Incompleteness they see does not shake their confidence in this stability, as the subject matter or nature of the underlying objects, or both, are of very great distance - in a fundamental sense - from that of the mathematics they are familiar with.

I initiated Concrete Mathematical Incompleteness (CMI) in the late 1960s with the idea that the future of Incompleteness depends on its success. Here we are 45 years later.

I discuss the continually evolving examples of CMI with many mathematicians, including top luminaries such as A. Connes, C. Fefferman, H. Furstenburg, T. Gowers, M. Gromov, D. Kazhdan, Y. Manin, B. Mazur, D. Mumford. All of these luminaries are fully aware of what is at stake, and had interesting reactions.

Above all, it is obvious that they do not identify, in any way, “fundamental or important mathematics” with 'mathematics people have done or are doing now”. My examples were judged strictly on fundamental mathematical standards of simplicity, naturalness, intrinsic interest, and depth. This is fortunate since the integration of CMI with existing concrete mathematical developments will likely occur only at later stages of CMI.

The outer limits of CMI live in the Borel measurable sets and functions between Polish spaces. New examples live in the finite sets of positive integers under addition only, and are explicitly Pi-0-1. The corresponding systems range from 2 quantifier induction to the huge cardinal hierarchy. Some historical highlights are: long finite sequences, continuous comparability of countable sets of reals, Kruskal's and Higman's theorem, Graph Minor Theorem, Borel Diagonalization, Borel determinacy, Borel selection, and Boolean Relation Theory.

The main focus of the talk will be on the current state of CMI.