OVERVIEW

The lecture course introduces to set theory in its axiomatic form, by explicitly listing basic axioms and, from those, deriving basic theorems which are of major relevance for the mathematical practice. Then the course will be devoted to the development of appropriate techniques to get to independence proofs.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to introduce the language of set theory, both as a foundational framework for mathematics and for its intrinsic interest. Students will learn the axioms of set theory along with initial developments and set-theoretic constructions, numeric sets, ordinal and cardinal arithmetic, principles of transfinite induction and recursion, the continuum problem, and elements of infinite combinatorics and the forcing method for independence proofs.

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 set theory as useful fonudation in the practice and didactics of mathematics, and presents the main tools for its applications. The course develops the theory and applies it to mathematical practice, also by means of examples from the students' previous experience.

PREREQUISITES

Proficiency in basic mathematical subjects and proofs.

TEACHING METHODS

Teaching style: In presence

SYLLABUS/CONTENT

The course will be about some of the following, depending on specific requests of the attendees:

Axiomatic presentations of set theory: ZF and NBG, ordinals, cardinals
The axiom of foundations, equivalent forms
The axiom of choice, equivalent forms
Cantor's and Dedekind's constructions of the real numbers, the continuum hypothesis
Models of ZF, relativization, relative consistency, constructible sets, inaccessible cardinals
Boolean-valued models, independence proofs

RECOMMENDED READING/BIBLIOGRAPHY

Elliott Mendelson, Introduzione alla logica matematica. Boringhieri 1975

Thomas Jech, Set Theory. The third millenium edition, Springer 2002

John Bell, Set Theory. Boolean-valued Models and Independence Proofs, Oxford University Press 1984

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

February 23rd, 2026

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of an oral examination on the arguments of the lecture course; it asks for the presentation of the subjects taught in the course and the solution of exercises.

ASSESSMENT METHODS

The exam verifies the actual acquisition of the mathematical knowledge of set theory and determines the skills developed to use such knowledge by means of problems and open questions. It aims at evaluating that the student has acquired an appropriate level of knowledge and analytical skills.

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

FURTHER INFORMATION

Ask professor for other information not included in the teaching schedule.