Enrollment year
2020/2021
Academic discipline
MAT/04 (COMPLEMENTARY MATHEMATICS)
Department
DEPARTMENT OF MATHEMATICS "FELICE CASORATI"
Curriculum
PERCORSO COMUNE
Period
2nd semester (01/03/2023 - 09/06/2023)
Lesson hours
48 lesson hours
Prerequisites
Sequences, numerical series, limits, classical numerical sets
Learning outcomes
The course aims to offer an analysis on the mathematical method, on the classical and modern axiomatic systems, on the meta-theoretical issues arisen in the 20th century, and on the attempts to solve the problem of foundations of mathematics.
Course contents
Short recap of set theory
Peano's Arithmetic: independence of axioms; definition by induction; addition, multiplication and order.
Cantorian set theory: comparing of infinite sets, countable and uncountable sets. Cantor's Theorem.
Paradoxes and crisis of foundations. Frege and the Russell's antinomy. Zermelo-Fraenkel set theory.
Equivalent formulations of the axiom of choice.
Construction of number sets: integer, rational, real numbers through Dedekind's cuts and through Cauchy's sequences.
Foundations of geometry. Birkhoff's metrical approach to the foundations of geometry
Teaching methods
Lectures and discussions on the theoretical part and on the solution of problems and exercises.
Reccomended or required readings
R.R. Stoll: "Set theory and logic", Dover.
J. Roitman: "Introduction to modern set theory", Wiley and Sons
K. Hrbacek, T Jech: "Introduction to set theory", Marcel Dekker
R.S. Millmann, G.D. Parker: "Geometry. A metric approach with models"
- Teacher's notes
Assessment methods
Written and oral examination, with the goal of evaluate the knowledge of the topics presented during the course. The access at the oral examination is possible only after a positive evaluation of the written examination.
Sustainable development goals - Agenda 2030