# Mathematische Logik 2

## Dr. Heinrich Rolletschek

### February 25, 2011

Thursday, 8:30-10:00, HS 11, beginning 10.3.2009

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

- 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. It will be dealt with relatively briefly in this lecture, since there is also a special
lecture covering this topic. (The lecture notes contain some further material.)
- 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.

Lecture notes will be handed out.

Oral exams will be given after the course.