Set theory: large cardinals, forcing, and their applications

Summary

ZFC set theory is the most widely used foundation for mathematics. With this standard framework precisely articulated, it is possible for us to explore what goes beyond it. Forcing is the standard technique used to show that various statements can neither be proven true nor proven false in ZFC. Similar statements are the so-called large cardinal axioms - these strengthen the theory in a meaningful way and so cannot be proven true, but also we do not expect them to be disprovable either. Both forcing and large cardinal axioms have been extremely useful, both for answering questions within set theory and for applications to other areas of mathematics. In this project, the student will advance the state-of-the-art working with these tools.

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