Set-Theoretic Hypodoxes and co-Russell's Paradox

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.