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Mathematical logic has been developed to get answers to fundamental questions such as

–  What is a mathematical proof?

–  Why can we trust in a formally correct mathematical proof?

In this way mathematical logic joins the field of mathematics and theoretical computer science to the philosophy of sciences. To point out these connections is the aim of the course.

The main topics of this course are so-called first-order languages. These are rather simple formal languages whose formulas can be interpreted as statements on mathematical objects and structures. Because of the purely formal character of such languages we can precisely define what it means that a formula

–  is a logical consequence of a set of other formulas

–  can be formally proved using a set of other formulas as premises

and we will show that for first-order languages the concepts of logical consequence and of formal provability are equivalent: This is Kurt Gödel’s famous completeness theorem.

Lectures at the blackboard / copies of the instructor’s script will be available

Oral examinations after the last lesson / dates by appointment

H.-D. Ebbinghaus / J. Flum / W. Thomas

Einführung in die mathematische Logik

Spektrum Akademischer Verlag 2007

Additional textbooks on Mathematical Logic will be announced during the course.

By first-order languages we can describe elementary algebraic and relational structures such as groups, rings, fields, ordered sets, lattices etc. But we can also regard these languages themselves as mathematical structures: Their terms and formulas are finite sequences of symbols that can be examined by means of discrete mathematics and set theory. Therefore some knowledge of elementary algebra, discrete mathematics and set theory (including cardinality of sets and the axiom of choice) is presupposed.

In this course we can only give an introduction to rather elementary parts of mathematical logic. In case of interest further courses on other parts of logic such as axiomatic set theory, theory of recursive functions, model theory etc. could follow.