Set-Theoretic Hypodoxes and co-Russell's Paradox
Timotej Šujan
TL;DR
The paper addresses the unclear notion of set-theoretic hypodoxes by focusing on the co-Russell set within a BST/$ extsf{NST}$ framework, examining independence issues across fragments and the role of paradoxical constructions. It develops a formal landscape of paradoxical sets and paradoxical groups, and then demonstrates a co-Russell-based paradox inside an NST fragment via a construction that yields mutually conflicting membership statements. It further extends the analysis with a Fixed Point/diagonalization approach, obtaining a diagonal contradiction in a non-UC setting through a Z^2 construction. The results suggest that, as currently formulated, there is no single property of independence that cleanly isolates hypodoxes, leaving the concept as an open, nuanced area requiring refined criteria and further study for a robust formalization.
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 Naïve Set Theory by exploiting the unique properties of the co-Russell set, relying on the Fixed Point Theorem of Naïve Set Theory. This result suggests that the boundary between paradoxes and hypodoxes may not be as clear-cut as one might assume.