Seminar: Set Theory and the Continuum Hypothesis, WS 2018/19

Dr. Holger Leuz (Institut für Philosophie) / Prof. Dr. Clara Löh / Johannes Witzig

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Seminar: Set Theory and the Continuum Hypothesis

The classical foundation of Mathematics consists of Logic and Set Theory. A popular formalisation of Set Theory is through the Zermelo-Fraenkel axioms (and the Axiom of Choice).

An innocent-looking assumption about basic set theory is that there is no set whose cardinality is strictly between the cardinality of the natural numbers and the real numbers (Continuum Hypothesis). In the early days of Set Theory, it was one of the fundamental open problems to determine whether the Continuum Hypothesis holds or not -- in fact, this is the first of the list of Hilbert's problems from 1900.

Surprisingly, it turns out that the Continuum Hypothesis (CH) is independent from ZFC, i.e., that one can neither prove nor disprove the Continuum Hypothesis from ZFC (!). Gödel proved in 1940 that CH cannot be disproved from ZFC; Cohen established in 1963 that CH also cannot be proved from ZFC (thus winning the Fields Medal in 1966).

In this seminar, we will develop the basics of Set Theory needed to properly formalise and prove these results. This result is also important for the epistemology of mathematics as it shows us certain limits for the classical foundation of mathematics and for the axiomatic method in general.

Time/Location

Mondays, 12:15 -- 13:45, M 103

Material

Prerequisites

None (except some maturity with formal reasoning).
This seminar is suitable for Bachelor/Master students and Lehramtsstudenten with an interest in set theory and foundations of mathematics, and could be the starting point for a project under my supervision (e.g., bachelor or master thesis).

Formalities/Credits

See the commented list of courses.

Last Change: October 15, 2018.