Modern Set Theory

A knowledge of modern set theory is essential for philosophers interested in the philosophy of mathematics as well as a powerful tool in mathematical philosophy. The formative years of the mathematical discipline of set theory are deeply connected with one of the most influential periods of mathematics in the last century, the so-called foundational crisis of mathematics (approx. 1900-1930). Set theory emerged in this period as an axiomatic theory and the question of completeness and consistency of such theories led to one of the most famous discoveries in the field of mathematical logic, Gödel’s incompleteness theorems. Nowadays, set theory provides us with powerful techniques (in particular the forcing technique) to explore mathematical independence phenomena and, more generally, provides the technical setup to research philosophical questions through mathematical means.

This workshop aims at introducing the mathematical background to understand set theory and the philosophical questions connected to it. It is structured into three sections: the axioms, the theory of infinite ordinals and cardinals, and set theoretic metamathematics. The first section explains set theory in the framework of the axiomatic system of ZFC: We introduce the axioms and explain their use and mathematical content; we further show how “normal” mathematics can be developed in this framework. The second section is concerned with the question of mathematical infinity: We define the hierarchy of transfinite ordinals and cardinals and their arithmetic; we look into the structure of the cumulative hierarchy and give an outlook on the question of large cardinals. In the last section we will take a look into advanced set theory that is concerned with the study of models of set theory. Here we will give an introduction to the forcing technique and how outer models of ZFC can be reached.

To deepen the theoretical knowledge presented in the workshop we will work on exercises during the afternoon sessions. To motivate the mathematical work we will also give an outlook on interesting philosophical questions that are arise out of the mathematical content.