2014/2015

2016/2017

DM270

MAT/04 (COMPLEMENTARY MATHEMATICS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

3°

2nd semester (01/03/2017 - 09/06/2017)

6

48 lesson hours

ITALIAN

ORAL TEST

ANTONINI SAMUELE (titolare) - 6 ECTS

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

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.

Axiomatic method and meta-theoretical issues. Examples of classical and modern axiomatic systems. Peano Arithmetic. Cantorian set theory. Paradoxes, crisis of foundations, foundational programmes. Zermelo-Fraenkel set theory. Construction of number sets (integer, rational and real numbers).

Extended summary

Axiomatic method: primitives 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.

Written and oral examination.