# 95669 - Logic (2) (Lm)

## Learning outcomes

At the end of the course, students are supposed to become acquainted with the semantics and the proof theory for classical logic or some extensions, such as modal logic, or alternative systems such as intuitionistic logic or other non classical logics.

## Course contents

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

What kind of relationships connect then syntactic theories with underlying mathematical structures? How do the properties satisfied by the earlier reflect in those fulfilled by the others? For instance, first order theories whose validity is preserved from models to their substructures can be axiomatized by using universal formulas only. Another example: mathematical theories satisfying certain algebraic properties admit the “quantifier elimination”, that is, every formula in their language is equivalent within the theory to some quantifier-free formula. If moreover a mechanical method exists to determine the truth of closed quantifier-free formulas, then the theory in question 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, such as the Mordell-Lang conjecture).

The purpose of this course is that of introducing the basic notions of Model Theory according to the following directions: Tarskian semantics for predicate logic; fundamental algebraic structures: the notions of monoids, groups, rings; Compactness Theorem and its consequences; universally axiomatized theories; categorical theories: linear dense orders without extrema and torsion-free Abelian groups; elimination of quantifiers: linear dense orders without extrema and algebraic closed fields; decidable theories: Vaught test, linear dense orders without extrema and algebraic closed fields; ultrafilters: their relations with the Compactness Theorem and the foundations of Non Standard Analysis.

Non-attending studets are referred to the instructions contained in the section Readings/Bibliography.

The fundamental didactic material will be the handouts provided by the teacher.

Non attending students must also study the chapters 1 and 2 of the handbook by A. Marcja, C. Toffalori: Introduzione alla teoria dei modelli, Pitagora, 1998.

## Teaching methods

Lessons in classroom with electronic blackboard also available in streaming.

## Assessment methods

The final exam will consist in an oral test, in which students are asked to prove their correct comprehension of the notions dealt with during the course, by oral explanation and also by written reconstruction of the fundamental definitions, results and proofs.

The final exam will consist in the exposition of a topic selected by the student and in another question chosen by the teacher.

## Teaching tools

- On line streaming

- Electronic blackboard

- Handouts provided by the teacher

## Office hours

See the website of Guido Gherardi