Set-Theoretic Hypodoxes and co-Russell's Paradox

TL;DR

This paper explores the ambiguous nature of set-theoretic hypodoxes, especially the co-Russell set, and demonstrates how contradictions can arise in naive set theory, challenging the clear distinction between paradoxes and hypodoxes.

Contribution

It clarifies the concept of hypodoxes in set theory and shows how contradictions can be derived using the properties of the co-Russell set.

Findings

Contradictions can be derived in naive set theory using the co-Russell set.

The boundary between paradoxes and hypodoxes is less clear than previously thought.

The Fixed Point Theorem plays a key role in deriving these contradictions.

Abstract

In this paper, we argue that while the concept of a set-theoretic paradox (or paradoxical set) can be relatively well-defined within a formal setting, the concept of a set-theoretic hypodox (or hypodoxical set) remains significantly less clear--especially if the self-membership assertion of the co-Russell set, $\{x:x\in x\}$, is classified as hypodoxical, whereas other set-theoretic sentences with no apparent connection to paradoxes are not. Furthermore, we demonstrate in detail how a contradiction can be derived in Na\"{\i}ve Set Theory by exploiting the unique properties of the co-Russell set, relying on the Fixed Point Theorem of Na\"{\i}ve Set Theory. This result suggests that the boundary between paradoxes and hypodoxes may not be as clear-cut as one might assume.