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.
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
- 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