Elementary set theory through the classical axiomatization by Zermelo and Fraenkel. The axiom of choice and its consequences. The theorem of Banach and Tarski. Mathematical paradoxes. Ruler-and-compass constructions. Paper folding constructions. Mathematical paradoxes. Algebraic curves that solve classical problems. Some simple facts in the theory of numbers.
M. Barlotti "Teoria degli insiemi" and C. Casolo "Appunti di teoria elementare dei numeri" – freely downloadable from the e-learning webpage for this course.
Learning Objectives
Knowledge acquired:
The axioms by Zermelo e Fraenkel. The theorem of Banach and Tarski. A characterization of the real numbers which can be constructed with ruler and compass. Some paper folding constructions.
Competence acquired:
The axiomatic construction of set theory. The theorem of Banach and Tarski. Ruler-and-compass constructions. Paper-folding constructions.
Skills acquired:
Constructing the number sets by axioms. Splitting a sphere in thirty pieces and reassembling them to build two isometric copies of the same sphere. Constructing numbers with ruler and compass. Trisecting an angle by paper folding. Using the method of coordinates to solve geometric problems and more generally mathematical problems.
Prerequisites
None. It is recommended to have passed the examinations of Algebra I and II and to have confidence with Cartesian coordinates in the plane.
Teaching Methods
Lectures.
Further information
Attendance of the lectures is not compulsory but is strongly recommended.
Type of Assessment
Oral examination on the program.
Course program
Elementary set theory through the classical axiomatization by Zermelo and Fraenkel. The axiom of choice and its consequences. The theorem of Banach and Tarski. Mathematical paradoxes. Ruler-and-compass constructions. Paper folding constructions. Mathematical paradoxes. Algebraic curves that solve classical problems. Some simple facts in the theory of numbers: multiplicative functions; Moebius' function, Euler's function, functions defined by the number and the sum of the positive divisors; perfect mumbers: congruences and equations: Hensel's lemma; quadratic residues; the quadratic reciprocity law; integers which are the sum of two, three and four squares; Waring's problem; Chebyshev's functions; Bertrand's postulate.