CODE  107014 

ACADEMIC YEAR  2021/2022 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/01 
TEACHING LOCATION 

SEMESTER  2° Semester 
TEACHING MATERIALS  AULAWEB 
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.
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.
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 metatheoretical study of logical systems, their applications, and the automation of proofs.
An introductory logic course
Traditional lectures
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.
SARA NEGRI (President)
RICCARDO CAMERLO
GIUSEPPE ROSOLINI (President Substitute)
All class schedules are posted on the EasyAcademy portal.
Valutazione basata su esercizi svolti durante il corso ed una presentazione finale su una tematica da concordare con la docente.