Ad.1. A model is a mathematical structure like the ring of integers Z or the ordered field of the reals R. Model theory selects a so-called language, mostly first order, to describe models. Zentral to this task is a truth definition which determines which sentences are true in a given model. We shall prove Goedel´s completeness theorem for first-order logic and exihibit some (immediate) mathematical consequences thereof.

Ad.2. The usual system of set theory is ZFC = ZF + AC, where AC is the so-called Axiom of Choice. I will develop ZFC to such an extent as to make it plausible that all of conventional mathematics can be formalized (and deduced) in ZFC. Some interesting facts about AC will also be presented.

Ad.3. Another name for recursion theory is computability theory. This is the theoretical or mathematical foundation of Computer Science. But I shall present recursion theory in a mathematics-oriented manner without going into Computer Science Proper (with the exception of so me remarks andhints).
Ad.4. In proof theory I will give complete proofs of Goedel´s two incompleteness theorems. Then, in closing the lecture, I will sketch Gentzen´s consistency proof for Peano Arithmetic.

Title:

Survey of Mathematical Logic

Credits: 5

Contents

Mathematical Logic (ML) can be divided into four parts:
1. model theoy,
2. set theory,
3. recursion theory,
4. proof theory.
These four parts are not disjoint but overlap rather substantially. It is therefore possible to divide ML differently, which is done by several authors.
In the lecture I will present from each part two or three main results and, of course, some accompanying side results.

Ad.1. A model is a mathematical structure like the ring of integers Z or the ordered field of the reals R. Model theory selects a so-called language, mostly first order, to describe models. Zentral to this task is a truth definition which determines which sentences are true in a given model. We shall prove Goedel´s completeness theorem for first-order logic and exihibit some (immediate) mathematical consequences thereof.

Ad.2. The usual system of set theory is ZFC = ZF + AC, where AC is the so-called Axiom of Choice. I will develop ZFC to such an extent as to make it plausible that all of conventional mathematics can be formalized (and deduced) in ZFC. Some interesting facts about AC will also be presented.

Ad.3. Another name for recursion theory is computability theory. This is the theoretical or mathematical foundation of Computer Science. But I shall present recursion theory in a mathematics-oriented manner without going into Computer Science Proper (with the exception of so me remarks andhints).
Ad.4. In proof theory I will give complete proofs of Goedel´s two incompleteness theorems. Then, in closing the lecture, I will sketch Gentzen´s consistency proof for Peano Arithmetic.

Literature

[1] J.R. Shoenfield : Mathematical Logic, 1967, [2] Jon Barwise : Handbook of Mathematical Logic, 1977, [3] Peter G. Hinman : Fundamentals of Mathematical Logic, 2005 [4] Wolfgang Rautenberg : A Concise Introduction to Mathematical Logic, 2010