Logic of paradoxes in classical set theories

TL;DR

This paper introduces the productivity principle, a logical truth that acts as a regulator for set theories, helping to understand and identify paradoxical classes and limitations within classical set theories like ZFC.

Contribution

It formulates the productivity principle as a logical regulator, providing a new framework to analyze paradoxes and limitations in classical set theories.

Findings

The productivity principle explains the logical bounds of set concepts.

It shows how paradoxical classes follow from the principle.

It suggests a new approach for developing consistent set theories.

Abstract

Set theoretical paradoxes have a common root -- lack of understanding of why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $\forall x \ x=x$. In this paper, we formulate a logical truth which we call the productivity principle. Bertrand Rusell was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two-valued $\in$ - language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of $ZFC$ set theory, the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory -- the cumulative cardinal theory of sets.