# MATHEMATICAL LOGIC

7 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA

OVERVIEW

The lecture course presents intuitions and mathematical results which relate to the developments in mathematical logic of the last 80 years. This allows to perform a deep analysis of mathematical practice. The explicit study of mathematical logic let the expert increase the understanding of the mathematical sciences and produces a fundamental basis for the presentation of mathematical themes and for the accretion of one's own mathematical intuition.

## AIMS AND CONTENT

LEARNING OUTCOMES

Study of first-order theories and their models, to analyze semantic issues, such as completeness and compact theorems, and syntactic questions such as the incompleteness theorems.

AIMS AND LEARNING OUTCOMES

At the end of the lecture course, a student has improved one's awareness of the mathematical facts and one's own understanding abilities of themes in mathematics in order to

- use them effectively to produce judgements autonomously;
- improve one's communication abilities in mathematics;
- strengthen one's power to learn and to analize mathematical themes.

The course considers logic as useful means in mathematical practice and in the didactics of mathematics, and presents the main tool for the mathematical study of logic: category theory. By this, the course develops the mathematics of deductive calculi and of formal logical theories, also by means of examples from the students' previous experience.

PREREQUISITES

None. Fluency with mathematical notations is useful.

Teaching methods

Teaching style: In presence

SYLLABUS/CONTENT

The lecture course will present and discuss the following subjects:

- Logic for the mathematical practice and the mathematical teaching.
- Category theory as the mathematical tool to structure the study of logical theories: categories, functors, natural transformations, adjunctions, indexed categories.
- Examples from logic: the propositional calculus, first order logic, higher order theories.
- Formal theories and the deduction calculi. Representation theorems; the completeness theorem for propositional theories and for first order theories.
- Type theory. The problem of consistency. The mathematical practice and type theory.

RECOMMENDED READING/BIBLIOGRAPHY

Course notes and slides presented during the lectures will be available on aula@web, complemented by other material. Notes taken at the lectures and the material on aul@web are enough in preparation for the exam. The books listed below are good references.

Mac Lane, S. Categories for the working mathematician. Springer-Verlag.

Mendelson, E. Introduction to mathematical logic. Chapman and Hall.

## TEACHERS AND EXAM BOARD

**Ricevimento:** by appointment

Exam Board

GIUSEPPE ROSOLINI (President)

SARA NEGRI

RICCARDO CAMERLO

## LESSONS

Teaching methods

Teaching style: In presence

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

The exam consists of a written essay and of an oral examination which can be taken in either order. The written essay is on the arguments of the lecture course and asks for the presentation of particular subjects taught in the course an the solution of exercises. The oral examination is a presentation and an open discussion fo subjects in the syllabus. The final mark determines how the two tests complement each other. The oral examination can be taken in itinere.

Assessment methods

The written essay verifies the actual acquisition of the mathematical knowledge of the basic notions of mathematical logic and determines the skills developed to use such knowledge in the analysis of mathematical theories by means of problems and open questions. The oral examination consists mainly in a presentation of some part of the syllabus and aims at evaluating that the student has acquired an appropriate level of knowledge and analytical skills.