2018/2019

2019/2020

DM270

MAT/04 (COMPLEMENTARY MATHEMATICS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

2nd semester (02/03/2020 - 09/06/2020)

6

48 lesson hours

Italian

WRITTEN AND ORAL TEST

ANTONINI SAMUELE (titolare) - 6 ECTS

Sequences, numerical series, limits, classical numerical sets

The course aims to offer an analysis on the mathematical method, on the classical and modern axiomatic systems, on the meta-theoretical issues arisen in the 20th century, and on the attempts to solve the problem of foundations of mathematics.

Axiomatic method: primitive notions and axioms. Meta-theoretical issues in modern axiomatic systems: consistency, independence, completeness.
Peano Arithmetic: independence of axioms; definition by induction; addition, multiplication and order.
Cantorian set theory: comparing of infinite sets, countable and uncountable sets. Cantor's Theorem.
Paradoxes and crisis of foundations. Frege and the Russell's antinomy. Foundational programmes: logicism, intuitionism, formalism.
Zermelo-Fraenkel set theory. Construction of number sets: integer, rational, real numbers through Dedekind's cuts and through Cauchy's sequences.

Lectures and discussions on the theoretical part and on the solution of problems and exercises.

- Borga, M., Palladino, D. oltre il mito della crisi: fondamenti e filosofia della matematica nel 20 secolo. Brescia, La scuola, 1997.

- Fiori, C., Invernizzi, S. Numeri reali. Pitagora, 1999.

- Teacher's notes

Written and oral examination, with the goal of evaluate the knowledge of the topics presented during the course. The access at the oral examination is possible only after a positive evaluation of the written examination.