20410613 - LM430-Logic and mathematical foundations

To acquire the basic notions of Zermelo-Fraenkel's axiomatic set theory and present some problems related to that theory.

Curriculum

teacher profile | teaching materials

Mutuazione: 20410613 LM430 - LOGICA E FONDAMENTI DELLA MATEMATICA in Matematica LM-40 TORTORA DE FALCO LORENZO

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. In case of health emergency due to COVID-19 the way lectures will be given will depend on the conditions. In such a potential framework all the possible efforts will be done to limit the unavoidable damages related to the necessity to switch to online courses and interaction during the lessons will be preserved as much as possible. The lessons will be broadcasted live and won't be recorded.

Type of evaluation

Written and/or oral exam, of a duration usually between 45 and 60 minutes. In case of health emergency due to COVID-19 the way exams will be held will depend on the conditions. In such a potential framework all the possible efforts will be done to limit the unavoidable damages related to the necessity to switch to online exams.

teacher profile | teaching materials

Mutuazione: 20410613 LM430 - LOGICA E FONDAMENTI DELLA MATEMATICA in Matematica LM-40 TORTORA DE FALCO LORENZO

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. In case of health emergency due to COVID-19 the way lectures will be given will depend on the conditions. In such a potential framework all the possible efforts will be done to limit the unavoidable damages related to the necessity to switch to online courses and interaction during the lessons will be preserved as much as possible. The lessons will be broadcasted live and won't be recorded.

Type of evaluation

Written and/or oral exam, of a duration usually between 45 and 60 minutes. In case of health emergency due to COVID-19 the way exams will be held will depend on the conditions. In such a potential framework all the possible efforts will be done to limit the unavoidable damages related to the necessity to switch to online exams.

teacher profile | teaching materials

Mutuazione: 20410613 LM430 - LOGICA E FONDAMENTI DELLA MATEMATICA in Matematica LM-40 TORTORA DE FALCO LORENZO

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. In case of health emergency due to COVID-19 the way lectures will be given will depend on the conditions. In such a potential framework all the possible efforts will be done to limit the unavoidable damages related to the necessity to switch to online courses and interaction during the lessons will be preserved as much as possible. The lessons will be broadcasted live and won't be recorded.

Type of evaluation

Written and/or oral exam, of a duration usually between 45 and 60 minutes. In case of health emergency due to COVID-19 the way exams will be held will depend on the conditions. In such a potential framework all the possible efforts will be done to limit the unavoidable damages related to the necessity to switch to online exams.