Università degli Studi di Pavia

Dipartimento di Matematica ''F. Casorati''

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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