Follow the Flow: sets, relations, and categories as special cases of functions with no domain

TL;DR

Flow Theory offers a novel framework where functions, without domains, unify sets, categories, and relations, providing a new foundation that avoids paradoxes and aligns with mathematical practice.

Contribution

The paper introduces Flow Theory, a new approach where all functions are domain-less, unifying various mathematical structures and avoiding classical paradoxes.

Findings

Flow Theory encompasses sets, categories, and relations as special cases of domain-less functions.

It avoids Russell's paradox without the Separation Scheme.

Provides a new perspective on hierarchies and duality in mathematics.

Abstract

We introduce, develop, and apply a new approach for dealing with the intuitive notion of function, called Flow Theory. Within our framework all functions are monadic and none of them has any domain. Sets, proper classes, categories, functors, and even relations are special cases of functions. In this sense, functions in Flow are not equivalent to functions in ZFC. Nevertheless, we prove both ZFC and Category Theory are naturally immersed within Flow. Besides, our framework provides major advantages as a language for axiomatization of standard mathematical and physical theories. Russell's paradox is avoided without any equivalent to the Separation Scheme. Hierarchies of sets are obtained without any equivalent to the Power Set Axiom. And a clear principle of duality emerges from Flow, in a way which was not anticipated neither by Category Theory nor by standard set theories. Besides, there seems to be within Flow an identification not only with the common practice of doing mathematics (which is usually quite different from the ways proposed by logicians), but even with the common practice of teaching this formal science.