Master's degree in Mathematics

Advanced course in foundations of mathematics

Course code
4S001104
Name of lecturers
Peter Michael Schuster,
Coordinator
Peter Michael Schuster
Number of ECTS credits allocated
6
Academic sector
MAT/01 - MATHEMATICAL LOGIC
Language of instruction
English
Period
II semestre dal Mar 2, 2020 al Jun 12, 2020.

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.

Syllabus

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
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

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.





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