Elements of Basic Mathematics 2

A.Y. 2020/2021
Overall hours
MAT/01 MAT/02 MAT/03 MAT/04 MAT/05 MAT/06 MAT/07 MAT/08 MAT/09
Learning objectives
The course is comprised of two parts. The first part is an introduction to the analysis and to the formalisation of mathematical reasoning. Starting from a series of case studies based on the students' experience in Analysis, Algebra, and Geometry, we will arrive at a discussion of the basic notions of contemporary mathematical logic. The second part puts to work the knowledge and the competence acquired in the first part to survey a number of answers that, in the course of time, have been given to the question: What are the foundations of Mathematics? Within this survey, special attention will be given to the theory of sets, both in its naive and in its formalised version.
Expected learning outcomes
By the end of the first part of the course students will be able to recognise, critically discuss, and use the main logico-mathematical tools employed in a given mathematical proof. By the end of the second part, students will have acquired basic knowledge of naive and formalised set theory. Further, students will have learned the essential features of a number of approaches to the foundation of mathematics that are alternative to the theory of sets.
Course syllabus and organization

Single session

Lesson period
Second semester
Classes can be attended in person. Classes will be available to students in streaming and will also be recorded for distance learning.
Course syllabus
First Part. What does it mean to prove a theorem? Proofs in slow motion: case studies from the B.Sc Degree in Mathematics. The formalisation of mathematical language. The formalisation of logical inference and proofs. Through the Looking-Glass: an introduction to syntax vs. semantics in mathematical practice. Why is it useful to formalise mathematics?

Second Part. What are the foundations of mathematics? Historical sketch: from the crisis in foundations at the beginning of the 20th century to the major limitative results of the 1930s. Sets: naive theory, formalisation. The theory of sets as a foundation of mathematics. A survey of a number of alternative approaches to foundations: constructivism, category theory.
Prerequisites for admission
No prerequisites.
Teaching methods
Blackboard, slides, handouts.
Teaching Resources
Course materials will be discussed at the beginning of the course.
Assessment methods and Criteria
Interview. The exam is an interview. The final assessment is either "Passed" or "Failed".
MAT/01 - MATHEMATICAL LOGIC - University credits: 0
MAT/02 - ALGEBRA - University credits: 0
MAT/03 - GEOMETRY - University credits: 0
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0
MAT/06 - PROBABILITY AND STATISTICS - University credits: 0
MAT/07 - MATHEMATICAL PHYSICS - University credits: 0
MAT/08 - NUMERICAL ANALYSIS - University credits: 0
MAT/09 - OPERATIONS RESEARCH - University credits: 0
Lessons: 27 hours
Professor: Marra Vincenzo