1 Time and place:

Thursday, 8:30–10:00, HS 11, beginning 7.3.2013

2 Prerequisites:

Familiarity with syntax und semantics of predicate logic, as presented in the course Mathematical Logic 1, is desirable, although the fundamentals will be briefly repeated.

3 Contents:

• Chapter 1 contains the basics of first-order predicate logic. This is partially a review from the lecture Mathematical Logic 1, but some conventions and one particular collection of rules of inference are also introduced.
• Chapter 2 centers on Gödel’s Completeness Theorem. Some further results are obtained either as a simple consequence (Compactness Theorem), or by inspection of the proof (Theorem of Löwenheim, Skolem and Tarski).
• Chapter 3 contains some further classical results like Craig’s Interpolation Theorem. They involve a refinement of the proof technique used for Gödel’s Completeness Theorem.
• Chapter 4 concerns Gödel’s Incompleteness Theorem, arguably the most famous result in Mathematical Logic altogether. The original proof is based on self-referential statements, but another approach, using results in computability theory, will also be sketched.
• Chapter 5 deals with elementary extensions, which leads to nonstandard models for various theories. Elementary chains are formed by repeated elementary extensions; they are applied in various proofs.

Some philosophical (epistomological) consequences, which result from limitations of the expressive and deductive power of first-order predicate logic, will also be discussed. Such limitations are shown primarily by Gödel’s Incompleteness Theorem, but also by the existence of various nonstandard models.

4 Literature:

Lecture notes will be handed out.

5 Assessment:

Oral exams will be given after the course.