Cardinal characteristics and large continuum

Georg Cantor`s "Continuum Hypothesis", the question about the size or "cardinality" of the real line, was the first on David Hilbert`s famous list of 23 problems that he proposed in 1900: is the cardinality of the real line the next cardinality after the cardinality of the natural numbers, or are there subsets of the real line which are neither countable nor equinumerous with the real line itself? This question naturally leads to the investigation of subsets (often quite pathological subsets) of the real line. In this project we propose to study and develop techniques for building set-theoretic universes (that is, mathematical structures that satisfy the set-theoretic axioms ZFC) in which subsets of the real line with predescribed properties (typical example: not Lebesgue-measurable, but of small cardinality) exist. The techniques considered are variants of the method of "iterated forcing"; we point out several issues that cannot be solved by the current methods and try develop new methods.

Mathematical logic entered the modern era through the work of Kurt Gödel, who established his famous Completeness and Incompleteness Theorems in Vienna in the 1930's. This project, focused on set theory, the area of logic that most interested Gödel in his later years.We were specifically interested in analysing subsets of the real line. Set theory started with Georg Cantor's discovery that there are many different kinds of infinity; the smallest possible infinity is the infinity of countable" sets, and Cantor showed that the set of real numbers (all real numbers, rational and irrational) describes a larger" infinity.But how much larger? Is it the next larger infinity? Or the one after that? Or perhaps we have to pass infinitely many infinities before we reach the infinity of the real numbers? Paul Cohen's forcing method showed that it is possible to construct mathematical universes which have so different characteristics that the answers to the above question can turn out differently in different universes.In this project we constructed several such universes in which there are many different infinities below the infinity of the real numbers; several of them are well-known infinities, for example infinities connected with notions from analysis, such as: which functions can be integrated?