Dathematics: A Meta-Isomorphic Version of “Standard” Mathematics Based on Proper Classes

Abstract

We show that the (typical) quantitative considerations about proper (as too big) and small classes are tangential facts regarding the consistency of Zermelo–Fraenkel set theory with Choice. Effectively, we will construct a first-order logic theory D-ZFC (dual theory of ZFC) strictly based on (a particular sub-collection of) proper classes with a corresponding special membership relation, such that ZFC and D-ZFC are meta-isomorphic frameworks (together with a more general dualization theorem). More specifically, for any standard formal definition, axiom, and theorem that can be described and deduced in ZFC, there exists a corresponding “dual” version in D-ZFC and vice versa. Finally, we prove the meta-fact that (standard) mathematics (i.e., theories grounded in ZFC) and dathematics (i.e., dual theories grounded in D-ZFC) are meta-isomorphic. This shows that proper classes are as suitable (primitive notion) as sets for building a foundational framework for mathematics. Finally, we describe the “singular” consequences that this meta-fact possesses regarding the cognitive dimension of ZFC.