Elementary Mathematics from an Advanced Standpoint 1

A.Y. 2021/2022
Overall hours
Learning objectives
The aim of this course is to provide an introduction to the axiomatic Zermelo-Fraenkel Set Theory. The notions of finite and infinite sets, natural numbers, ordinals and cardinals will be given and studied, together with the related arithmetics. Furthermore, various equivalent forms of The Axiom of Choice will be given, highlighting the importance of such an axiom from both a foundational and a practical point of view.
Expected learning outcomes
Acquisition of awareness of the need for a formal, rigorous and axiomatic theory of sets, in contrast to the naive set theory, usually taken as a basis for mathematics. Critical capacity to use axioms and comprehension of the role of paradoxes.
Course syllabus and organization

Single session

Lesson period
First semester
More specific information on the delivery modes of training activities for the academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.
Course syllabus
Introduction: Hints on syntax and first-order theories. Set theory
alphabet and language.
Abstract sets. Membership relation. Axiom of extensionality.
Axioms of the empty set, of pairing, of union and of power set.
Axiom schema of separation. Operations between sets. Problem of
the existence of a universal set. Relations and functions,
families and products of families of sets.
Cardinality. Cantor's Theorem. Cantor-Bernstein- Schröder Theorem.
Reflexive sets. Successor. Hereditary sets. Axiom of infinity. The
set ω of natural numbers.
The Principle of mathematical induction with the strong version,
and the well-ordering principle. Properties of the set ω.
Peano's axioms. Sum and product in ω.
Finite sets. Axiom of choice. Theorem of primitive recursion on ω. Principle of induction and recursion for well-ordered sets. Infinite sets. Comparison between well-ordered sets.
Ordinals (von Neumann). Comparison between ordinals.
Axiom schema of replacement. Well-ordered principle for ordinals.
Cardinals. Hints on the arithmetic of ordinals. Countable sets.
Equivalence of Axiom of choice- Trichotomy- Zorn's Lemma- Well-Ordering Principle- Tukey's Lemma- Tychonoff's theorem- Every infinite set has the same cardinality as its square- Every set can be endowed with a group structure.
Introduction of the ordered field of real numbers via Dedekind's sections. .
Prerequisites for admission
No specific prerequisites are requested.
Teaching methods
Frontal lesssons.
Teaching Resources
-P. R. Halmos, Teoria elementare degli insiemi, Feltrinelli, 1970.
-G. Lolli, Dagli insiemi ai numeri, Bollati Boringhieri,1994.
Assessment methods and Criteria
The final examination consists of an oral exam.
In the oral exam, the student will be required to illustrate results presented during the course ain order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Lessons: 42 hours
Professor: Mantovani Sandra
Educational website(s)
Thursday 12.45-14.15, by appointment
Studio 1019, I Floor, Dipartimento di Matematica, Via Saldini, 50