Does the nonexistence of $\in$-cycles imply the nonexistence of an infinite descending $\in$-sequence? We are working over the theory of (ZFC - {Foundation}). I know that, in (ZFC - {Foundation}), the nonexistence of an infinite descending $\in$-sequence implies Foundation. However, I am considering the theory $T$ that consists of (ZFC - {Foundation}), along with an infinite set of axioms that collectively say there are no $\in$-cycles. So for example, one axiom would be $\neg \exists x (x \in x)$, another would be $\neg \exists x \exists y (x \in y \in x)$, another would be $\neg \exists x \exists y \exists z(x \in y \in z \in x)$, and so on. Does the theory $T$ prove that there is no infinite descending $\in$-sequence? Or, is there a model of $T$ where there is an infinite descending $\in$-sequence?
 A: No, the nonexistence of cycles does not imply the nonexistence of descending chains.
We can build a model of $\mathsf{ZF-Foundation+NoCycles}$ in which there is a descending $\in$-sequence as follows. This is a bit tedious, but here is a sketch:
We define our model in ordinal-indexed stages. $A_0$ consists of infinitely many elements, $a_i$ ($i\in\mathbb{Z}$), with the relation $\varepsilon$ (to distinguish it from true set elementhood) defined as $a_i\varepsilon a_j$ iff $i<j$. Having defined $A_\alpha$ for an ordinal $\alpha$, we build $A_{\alpha+1}$ as follows:

*

*Let $$New_\alpha=\{X\subseteq A_\alpha: \forall x\in A_\alpha(X\not=\{y\in A_\alpha: y\varepsilon x\})\}$$ be the set of subsets of $A_\alpha$ not yet "named" by an element of $A_\alpha$. For example, we'll always have $A_\alpha\in New_\alpha$.


*We set $A_{\alpha+1}=A_\alpha\sqcup New_\alpha$, with the $\varepsilon$ relation extended in the obvious way. (The reason we only add in "new" sets is to preserve extensionality.)


*Finally, for limit $\lambda$ we let $A_\lambda$ be the direct limit of the $A_\alpha$s for $\alpha<\lambda$ (basically, "take unions").
By continuing through the ordinals, we get a model of $\mathsf{ZF-Foundation+NoCycles}$ in which there is an infinite descending $\in$-chain.
