Foundations of mathematics
Academic Year 2024/2025 - Teacher: MARIA FLAVIA MAMMANAExpected Learning Outcomes
The main objective of the course is to provide students the conceptual and operational tools to connect as much as possible what has been studied in previous courses. In particular, it aims to provide students with a logical approach to the organization of a mathematical theory with particular emphasis on geometry, arithmetic and set theory.
In particular, the course has the following objectives:
Knowledge and understanding: Know the foundational aspects of mathematics on the set theory, arithmetic, geometry.
Applying knowledge and understanding: Apply the axiomatic method to the construction of the natural numbers, and geometries
Making judgments: Make judgments about the quality of the proposed solution and evaluate its effectiveness. Acquiring critical skills in the areas of mathematics.
Communication skills : Ability to communicate their mathematical knowledge.
Learning skills : Using the knowledge gained to acquire new knowledge.
Course Structure
Lessons will take place in bi-weekly meetings. Active student participation will be required: lessons will be face-to-face and participative.
If the course is taught in a blended mode or at a distance, the necessary variations may be introduced with respect to what has been previously stated, in order to respect the syllabus
Information for students with disabilities and / or SLD
To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or compensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco.
Required Prerequisites
No pre-requisite is required. Knowledge of elements of Algebra is recommended.
Attendance of Lessons
Detailed Course Content
Textbook Information
Attilio Frajese e Lamberto Maccioni (a cura di), Gli Elementi di Euclide, UTET, Torino 1970
Sopra gli assiomi aritmetici, Bollettino dell'Accademia Gioenia Di Scienze Naturali in Catania, 1-2, 1908
M. Kline, Storia del pensiero matematico, Vol.1 e 2. Einaudi, 1999
Durante l'anno vengono forniti agli studenti appunti redatti dal docente contenenti gli argomenti trattati durante le lezioni frontali (su Studium).
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | The logical organisation of a mathematical theory: axiomatic theories; propositional calculus and Boolean algebra; predicative calculus. | Teacher's notes |
| 2 | Outlines of Foundations of Geometry: Euclid's ‘Elements’. | Attilio Frajese e Lamberto Maccioni (a cura di), Gli Elementi di Euclide, UTET, Torino 1970 |
| 3 | Fundamentals of arithmetic: Peano's axioms and Pieri's axioms | Sopra gli assiomi aritmetici, Bollettino dell'Accademia Gioenia Di Scienze Naturali in Catania, 1-2, 1908 |
| 4 | Subsequent extensions of the number concept | Teacher's notes |
| 5 | Mathematical infinity: the problem of infinity in Greek mathematics; the infinitesimal calculus; concept of an infinite set; Cantor's set theory; cardinality of the countable and the continuous; comparison of cardinality; paradoxes of set theory; axiomatic set theory; the axiom of choice; segments of a well-ordered set; Zermelo's theorem; propositions equivalent to the axiom of choice. | Teacher's notesM. Kline, Storia del pensiero matematico, Vol.1 e 2. Einaudi, 1999 |
Learning Assessment
Learning Assessment Procedures
The final examination consists of an oral test.
The examination of the learning may also be conducted electronically, should the conditions so require.
Examples of frequently asked questions and / or exercises
Axioms of continuity;
Numbers according to Pieri and according to Peano
Axiomatic theories