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.
Lectures and discussions on the theoretical part and on the solution of problems and exercises.
Assessment Methods
Oral examination, aiming to evaluate the knowledge of the topics presented during the course.
Texts
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 M.J. Greenberg: " Euclidean and non-Euclidean Geometries", Freman and Company R.S. Millmann, G.D. Parker: "Geometry. A metric approach with models" - Teacher's notes
Contents
Peano's Arithmetic: independence of axioms; definition by induction; addition, multiplication and order. Integers and rational numbers. Real numbers according to Dedekind and Cantor. Axioms of continuity of the real line. Cantorian set theory: comparing of infinite sets, countable and uncountable sets. Cantor's Theorem. Paradoxes and crisis of foundations. The Russell's antinomy. Zermelo-Fraenkel set theory. Introduction to the arithmetics of cardinal numbers ond order types. Well ordered sets. Equivalent formulations of the axiom of choice. Foundations of geometry. Hilbert's metrical approach to the foundations of geometry
