Using ZF set theory (without the axiom of foundation), is

$$ \{z: \neg(\exists u_1,...,u_n)((z \in u_1)\land (u_1 \in u_2) \land ... \land (u_n \in z))\}$$

A set for any n? This is an analogue of Russell's paradox for $\in$-loops of circumference n.

Any help with proving this would be great. Thanks!


The idea is the same. If it is a set $Z$ then either $Z\in Z$ or $Z\notin Z$.

  • If it is in $Z$ then we can easily construct a loop by picking $u_i=Z$ itself.
  • If it is not in $Z$, then there is such loop, $Z\in u_1\in\ldots u_n\in Z$, then this defines a loop for $u_n$, which is impossible.

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