Foundations of MATHEMATICS

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.

The course will take place tuwice a week. An active participation of the students is required: the lessons will be frontal and participated.

Logical organization of a mathematical theory: axiomatic theories; propositional calculus and Boolean algebra; predicate calculus. Fundamentals of Geometry: "Elements" of Euclid - Books I and II; the "Grundlagen der Geometrie" Hilbert; axioms of continuity and non-Archimedean geometry. Fundamentals of arithmetic: Axioms of Peano axioms and Pieri; Successive enlargements of the concept of number. Mathematical infinity: the problem of infinity in Greek mathematics; the calculus; concept of infinite set; Cantor's theory of sets; cardinality of a countable and continuous; comparison cardinality; paradoxes of set theory; axiomatic set theory; the axiom of choice; segments of a whole well-ordered; Zermelo's theorem; equivalent to the axiom of choice propositions. The formal theories: the phenomenon of paradoxes; hints on logicism, intuitionism, and formalism; formal theories of the 1st and 2nd order; I note on non-standard system of real numbers; limits of formalism.

Attilio Frajese e Lamberto Maccioni (a cura di), Gli Elementi di Euclide, UTET, Torino 1970

D. Hilbert (a cura di) Fondamenti della geometria, Franco Angeli, Milano 2012

Sopra gli assiomi aritmetici, Bollettino dell'Accademia Gioenia Di Scienze Naturali in Catania, 1-2, 1908

M. Kline, Storia della matematica

Throughout the year, students are given notes prepared by the teacher containing the topics treated during the frontal lessons (of Studium ) .