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.
Introduction to set theory as a foundation of mathematics.
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
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.
None, it may be useful to have proficiency in mathematical proofs.
Teaching style: In presence
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
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
Ricevimento: by appointment
Ricevimento: By appointment, via email or after class
GIUSEPPE ROSOLINI (President)
JACOPO EMMENEGGER
The class will start according to the academic calendar.
The exam consists of a written essay and of an oral examination which can be taken in either order. The written essay is on the arguments of the lecture course and asks for the presentation of particular subjects taught in the course and the solution of exercises. The oral examination is a presentation and an open discussion on subjects in the syllabus. The final mark determines how the two tests complement each other. The oral examination can be taken in itinere.
The written essay 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. The oral examination consists mainly in a presentation of some part of the syllabus and 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.
