Forcing and Consistency Results

University Of North Texas, Denton TX

Investigators

Abstract

Set theory is a field of mathematical logic which provides a solid foundation and analysis of the idea of infinity used in mathematics. Among the historically important applications of set theory are independence results, which are theorems which assert that certain statements can neither be proved nor disproved from the axioms of mathematics, or equivalently, that certain statements together with their negations are both consistent with such axioms. As an example, the continuum problem, which was posed by the founder of set theory Georg Cantor, asks how the size of the real number line compares with the infinite cardinalities. This problem was shown to be independent by the work of Gödel and Cohen. In the process, Cohen introduced forcing, which is a powerful technique for creating models of set theory which can be analyzed in terms of the combinatorial properties of partially ordered sets. The goals of the project are to make advances in the theory of forcing, and to prove consistency results in combinatorial set theory. The main topics of the project are side condition forcing and its application to proving independence results. For several decades, elementary substructures have been an indispensable tool in many combinatorial arguments in set theory. The method of side conditions involves including elementary substructures in the conditions of a forcing poset, in order to ensure cardinal preservation or stronger properties, such as properness and the approximation property. The project will build on previous work of Krueger in the theory of adequate sets, which is a style of side condition forcing which has resulted in a number of advances in the area. For example, adequate sets were used to construct a forcing poset for adding a club subset of the second uncountable cardinal with finite conditions, while preserving CH, solving an open problem of Friedman. We aim to develop general techniques in side condition forcing, and to apply these techniques to solve problems in combinatorial set theory, especially problems related to the square principle, the approachability ideal, forcing axioms, the theory of trees, ideals and filters, and stationary sets.