CODE 107014 2021/2022 5 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA MAT/01 GENOVA 2° Semester AULAWEB

## OVERVIEW

Proof theory  is the branch of mathematical logic aimed at the study of proofs intended as mathematical objects, usually presented as inductively generated tree structures.

Proof theory, in addition to being one of the pillars of the foundations of mathematics, together with model theory, axiomatic set theory, computability theory, and category theory, plays a primary role in the theory of programming languages.

## AIMS AND CONTENT

### LEARNING OUTCOMES

In this course we will study deductive systems for classical and intuitionistic logic and their extensions for mathematical theories.

In particular, we will study the relationship between axiomatic systems, of natural deduction and sequent calculus, with particular emphasis on their use for the search for proofs.

### AIMS AND LEARNING OUTCOMES

The purpose of the course is, on the one hand, to present the central results of proof  theory and, on the other hand,

to enable the students to autonomously define and use deductive systems with good properties for the meta-theoretical study of logical systems, their applications, and the automation of proofs.

### PREREQUISITES

An introductory logic course

### SYLLABUS/CONTENT

0. Introduction to proof theory

1.  From natural deduction to sequent calculus

2. An invertible classical calculus for classical propositional logic

3. Constructive reasoning

4. Intuitionistic sequent calculus

5. Admissibility of contraction and cut

6. Consequences of cut elimination

7. Completeness

8. Quantifiers in natural deduction and in sequent calculus

9. Completeness of classical predicate logic

10. Variants of sequent calculi

11. Structural proof analysis of axiomatic theories, with examples from algebra, geometry, lattice theory.

12. The constructive content of classical proofs

In addition to articles and notes that will be made available on AulaWeb, the following textbooks will be used:

S. Negri and J. von Plato, Structural Proof Theory, Cambridge University Press 2001.

S. Negri and J. von Plato, Proof Analysis, Cambridge University Press 2011.

A.S. Troelstra and H. Schwichtenberg, Basic Proof Theory, second edition. Cambridge University Press 2000.

## TEACHERS AND EXAM BOARD

### Exam Board

SARA NEGRI (President)

RICCARDO CAMERLO

GIUSEPPE ROSOLINI (President Substitute)

## LESSONS

### Class schedule

The timetable for this course is available here: Portale EasyAcademy

## EXAMS

### ASSESSMENT METHODS

Valutazione basata su esercizi svolti durante il corso ed una presentazione finale su una tematica da concordare con la docente.