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.
M. Barlotti “Complementi di Algebra” – 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 and (for what attains to classical problems solved by algebraic curves) seminars.
Further information
Attendance of the lectures is not compulsory but is strongly recommended.
Type of Assessment
For attending students, 50% of the vote will be assigned according to the seminarial work and 50% of the vote will be assigned after an oral examination on the remaining part of the program.
For non-attending students, the vote will be assigned after an oral examination on the whole 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.