Dathematics: A Meta-isomorphic Version of 'Standard' Mathematics based on Proper Classes

TL;DR

This paper introduces a dual framework called D-ZFC based on proper classes, demonstrating that standard mathematics and dathematics are meta-isomorphic, thus establishing proper classes as foundational as sets.

Contribution

It constructs a first-order logic theory D-ZFC based on proper classes and proves its meta-isomorphism with ZFC, offering a new foundational perspective.

Findings

ZFC and D-ZFC are meta-isomorphic frameworks

Proper classes can serve as primitive notions for foundations

Mathematics and dathematics are structurally equivalent

Abstract

We show that the (typical) quantitative considerations about proper (as too big) and small classes are just 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 (classic) mathematics (i.e. theories grounded on ZFC) and dathematics (i.e. dual theories grounded on D-ZFC) are meta-isomorphic. This shows that proper classes are as suitable (primitive notions) as sets for building a foundational framework for mathematics.