A paradox of Russell type (Q1312285)

The authors rediscovered the well-known paradox of grounded sets. A set \(y\) is said to be grounded if there is no denumerable sequence \(\langle x_ n\rangle\) such that \(y= x_ 0\) and \(x_{i+1}\in x_ i\) for \(i\geq 0\). If the class \(G\) of grounded sets is a set, then it is easy to prove that \(G\in G\Leftrightarrow G\not\in G\). The earliest publication of this paradox known to me is by \textit{Shen Yuting} [J. Symb. Logic 18, 114 (1953; Zbl 0053.029)]. The class \(G\) is a subclass of Russell's class \({\mathfrak R}=\{x: x\not\in x\}\). The authors note that, if there is a set \(X\) such that \(X\in X\), then the power set \({\mathfrak P}(X)\in {\mathfrak R}\) and \({\mathfrak P}(X)\not\in G\); hence, if \(\mathfrak R\) is not the entire universe, then \(G\) is a proper subclass of \(\mathfrak R\). The authors' attempt in Remark 3 to directly construct a set in \({\mathfrak R}- G\), namely \(Y_ 0= \{0,\{-1,\{-2,\{-3,\dots\}\}\}\}\), depends on making very special assumptions about the underlying set theory.