The course on Logical Theories 2 is part of the integrative activities of the CdS in Philosophical Sciences. At the end of the course students will have acquired the following skills: understanding of logical-mathematical theories; in-depth knowledge of topics and issues of logic as well as the ability to discuss the topics presented during the class.
The course aims at providing students with a basic knowledge of Zermelo-Fraenkel axiomatic set theory.
The course aims at providing students with a basic knowledge of Zermelo-Fraenkel axiomatic set theory.
teacher profile teaching materials
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, 2018Type 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
Oral exam, of a duration usually between 45 and 60 minutes.