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2nd Semester 2023/24: Alternative Foundations of Mathematics

Yurii Khomskii

The standard foundational system for mathematics is the Zermelo-Fraenkel axiomatization of set theory ZFC, in which all of classical mathematics can be formalized. However, many alternative foundational systems have been proposed as well: class theories (MK or NBG), New Foundations (NF), intuitionistic and constructive set theory (IZF/CZF), non-wellfounded set theory (ZFA), paraconsistent and paracomplete set theory, and many others. Such alternative foundations can be philosophically motivated (e.g., attempt at preserving naive comprehension), or arising naturally from mathematical applications (e.g., class forcing), sometimes both.

The aim of this project is for students to study and present one of these alternative foundations, in a seminar format. Some standard literature will be provided as a starting point, but students are also free to seek out other, more exotic papers and present about them if they want.


Student presentations in the last week of June (one presentation of roughly 1:30h per student). At the end, each participant submits a short written summary of their research.


Basic knowledge of axiomatic set theory and logic.


Pass/Fail based on seminar presentation and write-up.