Programme

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

Core Documentation

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Type of delivery of the course

The course includes Face-to-face lectures; Discussions with students and debates on the discussed topics; Attendance is not mandatory but strongly recommended.

Type of evaluation

Written and/or oral exam, of a duration usually between 45 and 60 minutes.

Programme

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

Core Documentation

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Type of delivery of the course

The course includes Face-to-face lectures; Discussions with students and debates on the discussed topics; Attendance is not mandatory but strongly recommended.

Type of evaluation

Written and/or oral exam, of a duration usually between 45 and 60 minutes.

Programme

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

Core Documentation

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Type of delivery of the course

The course includes Face-to-face lectures; Discussions with students and debates on the discussed topics; Attendance is not mandatory but strongly recommended.

Type of evaluation

Written and/or oral exam, of a duration usually between 45 and 60 minutes.