Elementary set theory in the classical axiomatization by Zermelo and Fraenkel. The axiom of choice and its consequences. The theorem of Banach and Tarski.
Ruler-and-compass constructions. Algebraic curves that solve classical problems.
Paper folding constructions.
Some simple facts in the theory of numbers.
Marco Barlotti - Appunti di Teoria degli insiemi;
Marco Barlotti - Appunti sulle Costruzioni geometriche;
Carlo Casolo - Appunti di Teoria elementare dei numeri;
These textbooks will be freely downloadable in pdf format from the e-learning webpage for this course:
https://e-l.unifi.it/course/view.php?id=5065
Learning Objectives
The course aims to increase the student’s grasp of some mathematical ideas which are particularly relevant for a future teacher.
At the end of the course the student should have acquired the following basic knowledges:
(1) knowledge of the fundamental axioms of set theory in a slightly revised version of the Zermelo-Fraenkel construction;
(2) knowledge of the axiom of choice and of the equivalent axioms known as "well-ordering principle" and "Zorn's lemma";
(3) knowledge of other axioms which are equivalent to the axiom of choice;
(4) confidence with the main consequences of accepting (or, alternatively, rejecting) the axiom of choice;
(5) confidence with the notion of "cardinality" and with comparison of cardinalities;
(6) knowledge of the difference between equisectionability and equidecomposability of geometric figures;
(7) knowledge of the theory of classical geometric constructions with ruler and compass;
(8) knowledge of the theory of geometric constructions by paper folding as stated by Humiyaki Huzita and Koshiro Hatori;
(9) knowledge of some subjects in number theory.
At the end of the course, moreover, the student should have acquired the following basic abilities:
(1) to state correctly, with an adequate language, the fundamental axioms of set theory;
(2) to build, via the axioms of set theory, the set of natural numbers, the set of integers and the set of rational numbers;
(3) to define in the set of natural numbers the usual order, the operations of sum and product, and the euclidean division;
(4) to apply Zorn's lemma in mathematical proofs;
(5) to motivate why any two equivalent polygons are equisectionable;
(6) to do simple geometric constructions with ruler and compass;
(7) to decide, in many instances, whether a given geometrical construction can be done using ruler and compass only;
(8) to do simple geometric constructions by paper folding;
(9) to tackle specific problems in number theory.
Prerequisites
Confidence with Cartesian coordinates in the plane. A basic knowledge of abstract algebra (as taught in the courses of Algebra I and Algebra II) is recommended.
Teaching Methods
The course consists in lectures in which the theory is presented, with a stress on elementary (but not easy) questions that can be proposed to high school pupils in order to stimulate their interest in mathematics.
Problems (some of which taken from math competitions) will be proposed during the lectures and subsequently discussed and solved.
During the lectures comments and questions from the attending students are encouraged.
Further information
The course assigns 9 CFUs and is expected to consist of 72 hours of lectures.
Attendance of lectures is not compulsory but strongly suggested. Hopefully, the textbooks available for free download through the dedicated e-learning webpage should be sufficient for an adequate preparation.
Prof. Marco Barlotti is contactable in room 2.13 on the second floor of the building in via delle Pandette 9 in Firenze, usually on wednesdays from 3:00 to 5:00 PM: please send in advance an email to <marco.barlotti@unifi.it>. Through email it is also possible to arrange a different place and/or a different time for the meeting.
Type of Assessment
In order to attain the credits for this course the student must undertake an oral examination, in which questions are posed and the student's answers are evaluated in order to verify his knowledge of
(1) the fundamental notions tought during the course (definitions)
(2) the interrelations among such notions (theorems)
and
(3) specific techniques used to prove such interrelations.
During the examination, simple exercises might also be proposed in application of the techniques learnt in the course.
Course program
Leibniz's axiom. Primitive words. The axiom of extension. The separation axiom scheme. The empty set. The axiom of couples. The axiom of regularity. The axiom of union. Intersection of sets. The axiom of subsets. Complement. De Morgan's laws. Ordered pairs. Families of sets.
The axiom of infinity. The set of natural numbers. Transitivity. Peano's axioms. Definitions by induction. Sum in the set of natural numbers. Product in the set of natural numbers. Ordering the set of natural numbers. The euclidean division in the set of natural numbers.
The set of integers. Rational numbers. Towards the real numbers: the side and the diagonal of a square are not commensurable. "Pi" is irrational.
The axiom of choice. Zorn's lemma. The well-ordering principle. Well-founded sets. Applications of the generalized induction principle to well-ordered sets. Further axioms which are equivalent to the axiom of choice. The axiom of choice and Lebesgue measure.
Cardinality. Finite sets and countable sets. Comparing cardinalities. Cardinality of the union and of the cartesian product.
Equisectionability in the euclidean plane. Some results on equisectionability. Equisectionability and area.
Actions of a group on a set. Equidecomposability in the euclidean space. The Banach-Schröder-Bernstein theorem. Free groups. A free group generated by two rotations of the sphere. The theorem of Banach and Tarski.
Ruler-and-compass constructions. The algebraic characterization of geometric constructions which are attainable using ruler and compass only. Ruler-and-compass construction of some regular polygons. Conics and ruler-and-compass constructions. Ruler-and-compass construction of the radical axis of two circles. The angle cannot be trisected in general, and the cube cannot be duplicate, using ruler and compass only.
Geometric constructions by paper folding: the rules of Humiyaki Huzita and Koshiro Hatori. How to trisect any angle and how to duplicate any cube by paper folding. Some regular polygons attainable by paper folding. Conics by paper folding.
Divisibility in the set of integers. Multiplicative functions. Theorems of Euler, Hensel and Gauss on congruences, with applications to diophantine equations.
An outline of the additive theory of integers. The theorem of Lagrange, Waring's problem, Scnirelman's method.
Questions concerning the sums of subsets of the set of integers. Remarks on the distribution of prime numbers.