Foundations of MATHEMATICS

Logical organization of a mathematical theory: axiomatic theories; propositional calculus and Boolean algebra; predicate calculus. Fundamentals of Geometry: "Elements" of Euclid; Non-Euclidean geometries; 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; transfinite cardinal numbers and ordinal numbers. 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

E. Agazzi, D. Palladino, Le geometrie non Euclidee e i fondamenti della geometria da un punto di vista elementare, Editrice la scuola, Brescia

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 ) .