Mathematics

ALESSANDRO GAMBINI

General aims: to acquire basic knowledge and skills in axiomatic set theory and to be able to apply them in various contexts, including teaching.

Specific aims:
Knowledge and understanding: The successful student will have acquired basic notions and results in mathematical logic: axioms and main results of the theory ZF; ordinal numbers; the axiom of choice; cardinal numbers; paradoxes in several areas of mathematics.

Applying knowledge and understanding: The successful student will be able to solve exercises and problems referring to the topics covered and to application in other mathematical areas. S/he will perform computations with ordinal numbers and cardinal numbers; s/he is familiar with mathematical translations of the notion of infinity. S/he will be able to apply her/his knowledge in an education context.

Critical and judgmental skills: The successful student will be familiar with mathematical rigor and formalism. S/he will have reflected on known mathematical contents; s/he knows how to tackle questions about the foundations of mathematics in a critical way. S/he will be able to discuss the role of intuition and rigor in teaching mathematics in different situations.

Communication skills: The successful student will be able to present subjects and arguments in the oral test, and to explain what s/he learned.

Learning skills: The acquired knowledge will allow to study more specialized subjects. The student will be motivated to extend the acquired knowledge.

General aims: to acquire basic knowledge and skills in axiomatic set theory and to be able to apply them in various contexts, including teaching.

Specific aims:

Knowledge and understanding: The successful student will have acquired basic notions and results in mathematical logic: axioms and main results of the theory ZF; ordinal numbers; the axiom of choice; cardinal numbers; paradoxes in several areas of mathematics.

Applying knowledge and understanding: The successful student will be able to solve exercises and problems referring to the topics covered and to application in other mathematical areas. S/he will perform computations with ordinal numbers and cardinal numbers; s/he is familiar with mathematical translations of the notion of infinity. S/he will be able to apply her/his knowledge in an education context.

Critical and judgmental skills: The successful student will be familiar with mathematical rigor and formalism. S/he will have reflected on known mathematical contents; s/he knows how to tackle questions about the foundations of mathematics in a critical way. S/he will be able to discuss the role of intuition and rigor in teaching mathematics in different situations.

Communication skills: The successful student will be able to present subjects and arguments in the oral test, and to explain what s/he learned.

Learning skills: The acquired knowledge will allow to study more specialized subjects. The student will be motivated to extend the acquired knowledge.

Basic algebra and analysis knowledge. Knowledge of intuitive set theory.

Paradoxes and antinomies
The intuitive set theory
The axioms of Zermelo and Zermelo-Fraenkel
Well-order and transitive sets
Mathematical induction
Ordinal numbers and ordinal arithmetic
The successions of Goodstein
Axiom of choice and equivalent statements
Cardinal numbers and cardinal arithmetic
The Continuum hypothesis

Dispense del corso: Fondamenti della Matematica
Claudio Bernardi, Mario Magnago, Marco Rainaldi, Mariella Serafini

D. van Dalen, H.C. Doets, H. de Swart, Sets: naive, axiomatic and applied, Pergamon Press, 1978

V.M. Abrusci, L. Tortora de Falco, Logica: Volume 2 - Incompletezza, teoria assiomatica degli insiemi, Springer 2018

A. Abian, La teoria degli insiemi e l'aritmetica transfinita, Feltrinelli, 1972

K. Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland Publishing Company, 1980

K. Hrbacek, T. Jech, Introduction to set theory, Dekker, 1999

T. Jech, Set theory, Springer, 2003

P. J. Cohen, Set theory and the continuum hypothesis, W.A. Benjamin, 1966

H. Rubin, J.E. Rubin, Equivalents of the axiom of choice, North-Holland, 1970

Interactive lectures with discussion.

Class attendance is not compulsory but recommended

Short written text followed by an oral examination to verify the knowledge and understanding of the topics covered in the course.

What are the Zermelo-Fraenkel Axioms?
How are operations with ordinals performed?
What is a countable set?
What is the Vitali set?
What are Goodstein sequence?
What does the Axiom of Choice say?
What is the continuum hypothesis?

• Introduzione al corso


• Paradossi e crisi dei fondamenti


• Origine della teoria degli insiemi


• Teorema di Cantor e assioma di estensionalità


• Assioma dell'insieme vuoto, assioma di isolamento, assioma della coppia, assioma dell'unione


• Coppia ordinata, assioma dell'unione, prodotto cartesiano


• Assioma dell'infinito e numeri naturali


• Principio di induzione, esercizi


• Assiomi di rimpiazzamento e fondazione


• Assiomi di Peano e relazioni ben fondate


• Insiemi transitivi, relazioni d'ordine, buon ordine


• Discesa infinita di Fermat e definizione di ordinale


• Classificazione di ordinali


• Induzione transfinita e somme di ordinali


• Moltiplicazione tra ordinali e proprietà


• Aritmetica ordinale ed elevamento a potenza


• Forma normale di Cantor, successioni di Goodstein deboli


• Teorema di Goodstein, Assioma di Scelta


• Insieme di Vitali


• Enunciati equivalenti all'assioma di scelta


• Applicazioni del Lemma do Zorn e tricotomia dei cardinali


• I numeri cardinali


• Aritmetica cardinale


• Ipotesi del continuo e Lemma di Konig