# AXIOMATIC SET THEORY

OVERVIEW

The course presents the basics of set theory, developing the theory from the axioms and investigating some of its interesting aspects, towards the introduction of the techniques for independence proofs.

## AIMS AND CONTENT

LEARNING OUTCOMES

The course is an introduction to the language and development of set theory, both as a foundational theory for mathematics and for intrinsic interest. The axioms of set theory are presented with their first consequences, together with the set theoretic constructions of number sets. Then ordinal and cardinal arithmetic are discussed, with the principles of transfinite induction and recursion. The continuum problem and some topics in infinite combinatorics are considered. Finally, the forcing method for independence proofs is introduced.

AIMS AND LEARNING OUTCOMES

Some goals of the course are: show how set theory constitutes a foundational theory in which the entire current mathematics can be developed; investingate some aspect of the theory of intrinsic interest; provide an introduction to independence proofs.

At the end of the course, the student is supposed to master the set theoretic techniques and be able to use autonomously constructions and arguments that are typical of the theory.

PREREQUISITES

All the necessary notions will be defined in the course.

Hpweverm familiarity with the topics presented in a course of Mathematical logic can be useful, as well as some acquaintance with basic topics in algebra, analysis, and topology.

Teaching methods

Classroom lectures.

During the course, several exercises will be proposed, in order to verify the understanding of the subject. Students are encouraged to submit their solutions, which will be checked and discusses. If the exercises are evaluated positively, they may contribute to the final evalution.

SYLLABUS/CONTENT

- Axioms of set theory

- First consequences of the axioms

- Set theoretic definitions of the common mathematical objects

- Ordinal and cardinal numbers, and their arithmetic

- Equivalents of the axiom of choice

- Real numbers and the continuum hypothesis

- Applications to topology and measure theory

- Infinite combinatorics

- Introduction to independence proofs

RECOMMENDED READING/BIBLIOGRAPHY

- K. Kunen, The foundations of mathematics, College Publications 2009.

- K. Kunen, Set theory, College Publications 2013.

- Detailed notes of the course, available on aulaweb page

## TEACHERS AND EXAM BOARD

Exam Board

RICCARDO CAMERLO (President)

SARA NEGRI

GIUSEPPE ROSOLINI (President Substitute)

## LESSONS

Teaching methods

Classroom lectures.

During the course, several exercises will be proposed, in order to verify the understanding of the subject. Students are encouraged to submit their solutions, which will be checked and discusses. If the exercises are evaluated positively, they may contribute to the final evalution.

## EXAMS

Exam description

Oral examination.

Assessment methods

Oral interview.

If the exercises given during the course have been submitted, the interview may begin by discussing some of these.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

31/05/2021 | 09:00 | GENOVA | Orale | |

18/06/2021 | 09:00 | GENOVA | Orale | |

07/07/2021 | 09:00 | GENOVA | Orale | |

30/08/2021 | 09:00 | GENOVA | Orale |