The continuum, forcing and large cardinals

The work shall study interactions between large cardinals and forcing and also with regard to properties of the continuum. The work shall be divided into three subtopics: (1) First subtopic is called "Combinatorial properties of cardinals and the continuum function". The focus shall be on the impact of several combinatorial properties on the continuum function (a function which maps kappa to the size of the powerset of kappa). As Easton showed, without extra assumptions, there is very little ZFC can prove about the continuum function on regulars. However, once we start to consider large cardinals, or in general combinatorial properties often formulated in the context of large cardinals (tree property, square principles, etc.), suddenly the situation is much more interesting. Subtopic 1 studies these connections. (2) Second subtopic is called "Combinatorial characteristics of the continuum and the Mathias forcing". This subtopic focuses on the real line from the point of certain combinatorial characteristics which shall be studied by means of the Mathias forcing. This forcing is determined by a filter on the powerset of omega. Thus it is natural to study the connections between the combinatorial properties of filters and the properties of the real line. This subtopic combines set theory and topology. (3) The third subtopic is called "New developments in the template iterations." Template iterations is a modern forcing method with the potential to show new results regarding the combinatorial characteristics of the real line. Among the main problems to be studied are: obtaining cardinal characteristics with singular values, application of template iterations on non-definable posets, and finally the application of the method with the mixed finite and countable support. These new approaches will be useful in obtaining new results.

This project explored several aspects of set theory, the study of infinity. We explored the behaviour of large infinities (also known as large cardinals), the combinatorial properties of orderings and graphs, as well as specific properties of the continuum (the real line). The project was carried out jointly by the Vienna group at the Kurt Goedel Research Center and the Prague group at the Logic Department of Charles University.