Advanced course in foundations of mathematics (2020/2021)

Course code
Name of lecturers
Peter Michael Schuster, Roberta Bonacina
Peter Michael Schuster
Number of ECTS credits allocated
Academic sector
Language of instruction
II semestre dal Mar 1, 2021 al Jun 11, 2021.

Lesson timetable

Go to lesson schedule

Learning outcomes

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.).

Reference books
Author Title Publisher Year ISBN Note
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
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
Shoenfield, Joseph R. Mathematical Logic. (Edizione 2) Association for Symbolic Logic & A K Peters 2001 1-56881-135-7
Peter Aczel, Michael Rathjen Notes on Constructive Set Theory 2010
Yiannis N. Moschovakis Notes on Set Theory Springer 1994 978-1-4757-4155-1
Peter Cameron Sets, Logic and Categories Springer 1998 978-1-4471-0589-3
Kenneth Kunen The Foundations of Mathematics (Edizione 2) College Publications 2012 978-1-904987-14-7

Assessment methods and criteria

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.

The assessment methods may be subject to change as the development of the situation demands. The distance modality will anyway be guaranteed for all students who require this in the academic year 2020-21.