Aspects of set theory with applications to relational structures
by
Katie Thompson
KGRC, University of Vienna

We will start out with a brief history of the foundations of set theory including the axioms of ZFC and models of set theory. A somewhat detailed explanation will be give on how to extend a model of set theory using forcing and contrast the outer models obtained in this way with inner models.

Why are we interested in expanding and shrinking models of set theory? The answer is decidability vs. undecidability. We can use forcing to show that certain mathematical statements cannot be decided (to be true or false) using only models a given axiomatic framework. What happens then if we add to our axioms? How strong does an axiom need to be to decide such statements? The notion of ßtrength" in this sense is a measure of how many questions an axiom can decide. Also, examples will be given where such axioms and other set theoretic assumptions were used as stepping stones or test cases in proving important mathematical theorems.

To give some flavor of how various aspects of set theory can be applied, we focus on the classification of relational structures. In particular, I am interested in sets with a single binary relation (with possible topological extensions), e.g. linear orders, partial orders, trees, graphs, directed graphs, boolean algebras. Some of these structures have been studied and classified to a certain extent using model theory. However, where model theoretic techniques fail, we can apply set theory to answer questions. The techniques used include infinite combinatorics, forcing and adding in axioms to decide questions.

Time permitting, we will also discuss an application for inner models and how this gives uniformized limitations to the power of forcing.

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Date received: July 15, 2008