1 Time and place:

Thursday, 8:30–10:00, HS 11, beginning 9.3.2017

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 deals with the basics of first-order predicate logic and is partially a review from the lecture Mathematical Logic 1. It contains various conventions and fundamental definitions and one particular collection of rules of inference.
• Chapter 2 centers on Gödel’s Completeness Theorem. 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 classical results like Craig’s Interpolation Theorem, whose proof involves a refinement of the 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 was based on self-referential statements, but another approach, using results in computability theory, will also be sketched.
• Chapter 5 deals with elementary extensions, leading to nonstandard models for various theories. Elementary chains, which are formed by repeated elementary extensions, 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. They become apparent most clearly by Gödel’s Incompleteness Theorem, but also by the existence of nonstandard models.

4 Literature:

Lecture notes will be handed out.

5 Assessment:

Oral exams will be given after the course.