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 course aims to introduce the foundational tools for the mathematical study of first-order theories and their models. Topics include completeness and compactness theorems, Gödel’s incompleteness theorems for arithmetic, and the paradigm of mechanical computation as realized by Turing machines.

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 their personal work, the students need 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 course syllabus includes the presentation and discussion of the following topics:

• Examples of the use of logic in mathematical practice.
• First-order theories: deductive calculations, theories, models, completeness theorem.
• Foundations of computability theory and incompleteness theorems.

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

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 an oral 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 will contribute to the final evaluation in the measure of 50%.

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.

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it