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Tree-Forcing Notions

Giorgio Laguzzi

Donnerstag, 2019-10-24 17:00Uhr

During the 1960s Cohen and Solovay introduced and developed the method of forcing, which soon became a key technique for building various models of set theory. In particular such a method was crucial for answering questions concerning the use of the axiom of choice to construct non-regular objects (such as non-Lebesgue measurable sets, non-Baire sets, ultrafilters) and to analyse possible sizes of several types of subsets of reals (such as dominating and unbounded families, and other so-called cardinal characteristics). One of the key ideas in both cases is the notion of a tree-forcing, i.e. a partial order consisting of a specific kind of perfect trees. In this talk, after a brief historical background, we will focus on some results on Silver, Miller and Mathias trees. We will also see applications of infinitary combinatorics and tree-forcing in the context of generalized descriptive set theory and the study of social welfare relations on infinite utility streams.