## Mathematische Logik 2

Dr. Heinrich Rolletschek

February 23, 2015

### 1 Time and place:

Thursday, 8:30–10:00, HS 11, beginning 5.3.2015

### 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.