Winter 2022/23

Set Theory and Model Theory (in englischer Sprache)

03-IMAT-STMT

Vorlesung
ECTS: 6

Termine:
w?chentlich Di 08:00 - 10:00 MZH 5600 Vorlesung Pr?senz
w?chentlich Di 14:00 - 16:00 MZH 5600 ?bung Pr?senz

Profil: SQ
Schwerpunkt: IMVT-SQ

Set theory and model theory

Intuitively, a set is a collection of all elements that satisfy a certain property. This intuition, however, is false! The following example is known as Russell's Paradox. Consider the set S whose elements are exactly those that are not members of themselves: S = { X : X is not element of X }. Is S an element of S? If S is an element of S, then S is not an element of S. On the other hand, if S is not an element of S, then S belongs to S. In either case we have a contradiction. We must revise our intuitive notion of a set. In the first part of the lecture we develop axiomatic set theory (ZFC) in the framework of first-order logic, which forms the foundation of modern mathematics. We cover the axioms of set theory, ordinal numbers and induction and recursion over well-founded relations, cardinal numbers and the axiom of choice.

In the second part of the lecture we turn to classical topics of first-order model theory. Model theory studies classes of mathematical structures, such as groups, fields, or graphs, from the point of view of mathematical logic. Many notions, such as homomorphisms, substructures, or free structures, that are commonly studied in specific fields of mathematics are unified by the general approach of model theory. We study ways to construct models with desired properties from first-order theories and the expressive power of first-order logic.