Math 6010 Spring 2018

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We will begin the course with an introduction to descriptive set theory. We will begin by following some course notes which are given below. This will develop the basic theory of Polish spaces and the main definability hierarchies considered for these spaces. At the beginning, the subject has a strong overlap with general topology/real analysis, but as we move along the subject takes on a distinct character. In particular, logic and set theory begin to enter into the subject in a significant way. Methods from both logic, recusion theory, and set theory play increasingly important roles as the subject develops.

After presenting the basic notions, we study the Borel sets, showing ``classical'' results such as every Borel set is either countable of size \( 2^\omega\) and connecting the \( {\boldsymbol{\Delta}}^0_\alpha \) sets with the difference hierarchy. We progress past the Borel sets, defining the analytic and co-analytic sets, and more generally the projective sets. We develop the basic theory of analytic and co-analytic sets from the modern perspective of scales and Suslin representations. We show how bsic methods from logic and set theory can be used to obtain results about these sets. We show that as we progress past this level the main questions start becoming independent of \( \mathsf{ZFC} \), and thus we must introduce large cardinal/determinacy axioms to progress. As time permits, we study additional topics in determinacy theory or other areas of descriptive set theory.

Notes on descriptive set theory: Polish spaces.

Notes on determinacy: Determinacy.

Good additional references are "Classical Descriptive Set theory" by A. S. Kechris, and "Descriptive Set Theory" by Y. N. Moschovakis.