77988 - Foundations of Mathematics (1) (LM)

Learning outcomes

In the end of the course students will have gained a critic view on the contemporary development on the foundations of mathematics (incompleteness of axiomatic systems and independence), that beside the traditional importance for didactics of mathematics is nowadays strictly connected to logical research and to theoretical computability theory. 

Course contents

Model Theory is a branch of Logic concerning the relationships joining logical language and mathematical structures. More precisely, it uses logical language as a profitable conceptual tool for “doing mathematics”. Logicians, in fact, do not use rulers or compasses, but rather the language itself, by which they construct theories of models they want to investigate.

What kind of relationships connect therefore syntactic theories with described mathematical structures? How do the properties satisfied by the earlier reflect in those of the latter, and vice versa? For instance, first order theories satisfying a certain precise relation with their models and their substructures can be axiomatized by using universal formulas only. Another example: mathematical theories satisfying other kind of properties with respect to the nature of their sub-models admit the “quantifier elimination”, that is, every closed formula in their language is equivalent within the theory to some closed quantifier-free formula (when this applies, the theory is decidable). Model Theory is hence a branch of Logic whose results can increase considerably our knowledge of mathematical structures (and remarkable applications have in fact occurred in algebraic geometry for the solution of important conjectures).

Purpose of this course is that of introducing the basic notions of Model Theory according to the following directions: Tarskian semantics for predicate logic; Compactness Theorem and its consequences; universally quantified theories; preservation of formulas in model transformations; decidable mathematical theories; elimination of quantifiers; definable sets.

As for the necessary theoretical prerequisites, ordinary school algebraic basic notions suffice together with the usual logical background provided by any first level Logic course.

Readings/Bibliography

The fundamental didactic material will be provided by the teacher. As side support material the following books are recommended:

A. Marcja, C. Toffalori: Introduzione alla teoria dei modelli, Pitagora, 1998

D. Marker: Model theory: an introduction, Springer, 2002

Teaching methods

Lessons in classrooms with blackboard.

Assessment methods

The final exam will consist in an oral test, in which students are supposed to prove their correct comprehension of basic notions and results of Model Theory.

Teaching tools

Lessons will be held by using a blackboard.

Typewritten didactic material will be provided by the teacher.

Office hours

See the website of Guido Gherardi