- 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. 5827)
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from Sep 22, 2022 to Dec 22, 2022
Learning outcomes
Please 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. Some special axioms: Regularity, Choice, Continuum 'Hypothesis'. Alternative theories: classes and NBG, nonstandard set theories. Hints on independence problems.
3. Ordinal and cardinal numbers: Cantor's approach to ordinality and cardinality. Von Neumann's approach. Burali-Forti's paradox. Ordinal arithmetics vs Cardinal arithmetics.
Fur further details, please read the italian section.
Readings/Bibliography
The books by MENDELSON and HALMOS are translations into Italian from English: the latter edition is also available in Department library. For further details, please read the related section in Italian.
Teaching methods
Please read the related section in Italian
Assessment methods
Please read the related section in Italian
Teaching tools
Please read the related section in Italian
Links to further information
Office hours
See the website of Piero Plazzi