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### Foundations of Mathematics

Professors:
Antonini Samuele
Year:
2015/2016
Course code:
500337
ECTS:
6
SSD:
MAT/04
DM:
270/04
Lessons:
56
Period:
II semester
Language:
Italian

#### Objectives

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 solving the problem of foundations of mathematics.

#### Teaching methods

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

#### Examination

Written and oral examination.

#### Prerequisites

Principal properties of: sequences, numerical series, limits, classical numerical sets (natural, integers, rational, real and complex numbers).

#### Syllabus

Axiomatic method: primitives and axioms. Meta-theoretical issues in modern axiomatic systems: consistency, independence, completeness. Euclidean Geometry and Hilbert Geometry.
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.

#### Bibliography

- 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

Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602