22 April 2022, Cool Logic, Valentin Müller

Among the variety of possible solutions to the paradoxes of naive set theory, suggestions made by the mathematician Heinrich Behmann (1891--1970) appear to be particularly remarkable (even though they are commonly unknown today). From Behmann’s point of view, the paradoxes do not represent proper contradictions, but rather “meaningless” expressions that can be avoided by a simple and purely syntactical criterion. The main goal of my talk is to provide a partial confirmation of Behmann’s view. To this end, I will present a new system of natural deduction strongly inspired by Behmann’s analysis of the set-theoretical paradoxes. It will be demonstrated that a certain subclass of the proofs in our system has the normalization property: every deduction in this class may be transformed into a “cut-free” proof. As a corollary, it then follows that the propositional fragment of our system is in fact consistent. In the last part of the talk, I will discuss some open problems and closely related approaches such as the system of “Fitch-Prawitz Set Theory”.