34756 - Foundations of mathematics

Academic Year 2017/2018

  • Docente: Piero Plazzi
  • Credits: 6
  • SSD: MAT/01
  • Language: Italian
  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Mathematics (cod. 8208)

Learning outcomes

Read the related section in Italian.

Course contents

Some knowledge about predicative and propositional logic is strongly recommmended.

1. Algorithms, arithmetic and Gödel incompleteness results. Algorithms. Turing machines. Recursivity and computation of arithmetical functions: primitive recursive functions, μ-recursivity. Recursive relations, enumerability. Church-Turing's thesis. Peano Arithmetics. Gödelization  and Gödel's Incompleteness Theorems.
2. Axiomatic set theory. Historical introduction: Cantor's early theorems on numerical sets, intuitive set theory and paradoxes (Cantor's and Russel's). The axiomatic approach: Zermelo-Fraenkel Theory ZF. Ordinal and cardinal numbers: von Neumann's approach. Some special axioms: Regularity, Choice, Continuum 'Hypothesis'. Alternative theories: classes and NBG, nonstandard set theories. Hints on independence problems.

Fur further details, please read the above section.

Readings/Bibliography

See the related section in Italian. The books by MENDELSON and  HALMOS are translations into Italian from English.

Teaching methods

Read the related section in Italian.

Assessment methods

See the related section in Italian.

Teaching tools

See the related section in Italian.

Office hours

See the website of Piero Plazzi