The Helsinki Logic Group
Mathematical logic uses exact mathematical methods, originally developed in algebra, topology, measure theory, analysis, and combinatorics to study the two thousand year old subject of logic. During the 20th century, thanks to the revolutionary results of Gödel, but also of Skolem, Gentzen, Church, Turing and Cohen, mathematical logic developed into a deep research area with applications to philosophy, computer science, linguistics and, indeed, mathematics itself. In mathematical logic the Helsinki Logic group focuses on set theory, set-theoretic model theory, model theory, finite model theory, dependence and independence logic, second order logic, philosophical logic, as well as the history of logic and foundations and philosophy of mathematics. The group has developed methods in infinitary logic involving transfinite games and trees to investigate the structure of uncountable models, with connections to stability theory. The group is also known for its work in generalized quantifiers: their hierarchies, their applications in linguistics and computer science, and their set-theoretical properties, as well as for its work in the theory of abstract elementary classes and metric model theory. A recent topic of interest is dependence logic, a project to develop the mathematics and logic of dependence and independence concepts, as they are used in mathematics, computer science and elsewhere.
In this course, we cover the basic theory of forcing, a basic method
of constructing a model of set theory, and we use it to prove the
consistency of ZFC + "the negation of the Continuum Hypothesis"
assuming the consistency of ZFC. If time permits, we also show the
consistency of ZF + "the negation of the Axiom of Choice" assuming the
consistency of ZFC and cover some applications of forcing.
During the course, we assume the familiarity of the basics of ordinals
& cardinals, and of the transfinite induction & recursion. We also
mention some basic facts about the Gödel's constructible universe.