In recent work with Emanuele Bottazzi, we introduced a notion of integration that allows for infinitesimal measures without needing the axiom of choice. The range of values of integrals of real functions is in general not a field but a “comparison ring”, which is preserved under reduced powers. This notion of integral is able to represent classical integrals in a canonical way. As an application, we define a geometric measure over an infinite-dimensional vector space that overcomes some of the well-known limitations for real-valued measures and addresses an old paradox about conditional probability. This leads to a notion of fractal dimension, and we apply Martin’s Axiom to explore the possible order structure among these dimensions.

This talk will be given in person.

Please be aware of the fact that you may be required to show proof of your 3G status upon entry of the buildings, or during sporadic random checks in the seminar rooms. During the Logic Colloquium we will also pass around an attendance sheet to facilitate contact tracing. (According to the regulations, this form will be kept for 28 days and destroyed thereafter.)