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 the mathematical practice. The explicit study of mathematical logic lets 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

The basic tools for the mathematical study of first order theories and their models are provided, and the completeness and compactness theorems and  incompleteness of arithmetic will be proved. Fundamental to this is the mechanical calculation paradigm implemented by Turing machines, which allows us to solve Hilbert's decision problem.

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 the practice, the didactics, and the research in mathematics, and presents the tools for the mathematical study of logic;  by these tools, 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

During lectures, the instructor explains the theory and its applications to several examples and for the resolution of the exercises. In his personal work, the student needs to acquire the knowledge and the concepts of mathematical logic, and be able to solve the exercises that will be assigned and discussed in class.

SYLLABUS/CONTENT

The lecture course will present and discuss the following subjects:

• Logic for the mathematical practice. Two instances: the elementary theory of sets, and the elementary theory of categories.
• First order theories: deductive calculi, theories, models, the Completeness Theorem.
• Computability: Turing machines, the Universal Macchina Theorem, the s-m-n Theorem.

RECOMMENDED READING/BIBLIOGRAPHY

The teaching material will be made available on Aulaweb, complemented by other material. Notes taken at the lectures and the material on Aulaweb are enough in preparation for the exam. The books listed below are good references.

Abrusci, V.M. & Tortora de Falco, L., Logica. Volume 1  Dimostrazioni e modelli al primo ordine, Springer, 2015.

Cantini, A. & Minari, P., Introduzione alla logica: linguaggio, signicato, argomentazione, Le Monnier, 2009.

Masini, A., In Viaggio con la Logica Simbolica, McGraw-Hill, 2023.

von Plato, J. Elements of Logical Reasoning, Cambridge University Press, 2013.

Schwichtenberg, H., Mathematical Logic (lecture notes), 2012.

Shoenfield, J.R., Mathematical Logic, Association for Symbolic Logic & A K Peters, 2001.

van Dalen, D., Logic and Structure, Springer, 2013.

TEACHERS AND EXAM BOARD

Exam Board

SARA NEGRI (President)

ALESSIO CAMINATA

JACOPO EMMENEGGER

ALDO CONCA (President Substitute)

EMANUELA DE NEGRI (President Substitute)

ALESSANDRO DE STEFANI (President Substitute)

STEFANO VIGNI (President Substitute)

ANNA MARIA BIGATTI (Substitute)

RICCARDO CAMERLO (Substitute)

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written examination. The exams is on the topics 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 solution of the exercises assigned during the lessons can contribute to the final evaluation.

Students with DSA certification (specific learning disabilities), disability or other special educational needs are advised to contact the teachers at the beginning of the course to establish teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.

ASSESSMENT METHODS

The exam verifies the actual acquisition of the mathematical knowledge of the basic notions of mathematical logic and evaluates the skills developed to use such knowledge in the analysis of mathematical theories by means of problems and open questions. It aims at evaluating that the student has acquired an appropriate level of knowledge and analytical skills. The evaluation takes into account the correctedness of the solutions, the clarity of the exposition, and the rigour of the arguments developed.

Exam schedule