## Mathematische Logik 2

Dr. Heinrich Rolletschek

February 16, 2017

### 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.