The Notion "Pathology" in Set Theory

TL;DR

This paper explores the concept of 'pathology' in set theory, distinguishing it from antinomies, and formalizes this notion to develop a system with a maximal universe.

Contribution

It formalizes the notion of set-theoretic pathology and introduces the system NACT* with a unique maximal universe based on hereditary non-pathological formulas.

Findings

Formal definition of pathology in set theory

Development of the NACT* system with a maximal universe

Comparison between pathologies and antinomies

Abstract

When we study the paradoxes of set theory we find out that there are mainly 2 types: the pathologies and the antinomies. These 2 notions are made precise and compared with the somehow inductively definable concept "abnormal". (See my paper "Naive Axiomatic Mengenlehre for Experiments" in arXiv.) In the following 5 Patho Theses are discussed in order to formalize this notion of pathology. This allows us to define formally the property "Hereditary-non-Pathological" for well-formed formulas. With this property the system NACT* of Naive Axiomatic Class Theory is constructed, which has a "unique maximal" universe (in a special sense).