This monographic course introduces advanced topics in the area of the foundations of mathematics and discusses their repercussions in mathematical practice. The specific arguments are detailed in the programme. At the end of this course the student will know advanced topics related to the foundations of mathematics. The student will be able to reflect upon their interactions with other disciplines of mathematics and beyond; to produce rigorous argumentations and proofs; and to read related articles and monographs, including advanced ones.
Introduction to Zermelo-Fraenkel style axiomatic set theory, with attention to constructive aspects and transfinite methods (ordinal numbers, axiom of choice, etc.).
Gödel's incompleteness theorems and their repercussion on Hilbert's programme, with elements of computability theory (recursive functions and predicates, etc.).
|Peter Smith||An Introduction to Gödel's Theorems (Edizione 2)||Cambridge University Press||2013||9781107606753|
|Torkel Franzén||Gödel's Theorem: An Incomplete Guide to its Use and Abuse.||A K Peters, Ltd.||2005||1-56881-238-8|
|Jon Barwise (ed.)||Handbook of Mathematical Logic||North-Holland||1977||0-444-86388-5|
|Riccardo Bruni||Kurt Gödel, un profilo.||Carocci||2015||9788843075133|
|Abrusci, Vito Michele & Tortora de Falco, Lorenzo||Logica. Volume 2 - Incompletezza, teoria assiomatica degli insiemi.||Springer||2018||978-88-470-3967-4|
|Peter Aczel, Michael Rathjen||Notes on Constructive Set Theory||2010|
|Yiannis N. Moschovakis||Notes on Set Theory||Springer||1994||978-1-4757-4155-1|
Single oral exam with open questions and grades out of 30. The exam modalities are equal for attending and non-attending students.
The exam's objective is to verify the full maturity about proof techniques and the ability to read and comprehend advanced arguments of the foundations of mathematics.