The following theory contains a comprehension axiom that is a naive like schema. This theory definitely looks inconsistent at first glance. However, I tried to find this inconsistency, but to no avail. The main idea pivotes around an *acyclic* membership relation $\in^*$, so an acyclic member of a set is an element of a set that doesn't contain that set in its transitive closure. Now, this theory allows free construction of sets after all formulas in the language FOL(=, $\in^*$). Formal workup: Language: the first order language of set theory. **Extesnionality:** $\forall x \, (x \in A \leftrightarrow x \in B) \to A=B$ **Transitive Closures:** $\forall x \exists t: t=TC(x)$ ***Define:*** $t=TC(x) \iff trs(t) \land x \subseteq t \land \forall k (trs(k) \land x \subseteq k \to t \subseteq k)$ Where "$trs$" stands for "is transitive", that is: closure under relation $\in$ **Induction:** if $\phi$ is formula, then: $$\forall y \in x \ (\phi) \land \forall k \, (\phi(k) \to \forall l \in k (\phi(l))) \to \\ \forall m \in TC(x)( \phi(m))$$ ***Define:*** $y \in^* x \iff y \in x \land \neg \, x \in TC(y)$ **Comprehension:** $\exists x \forall y \, (y \in x \iff \phi^*)$ Where $\phi^*$ is a formula not using $``x"$, whose predicates are among $=, \in^*$ symbols. Questions: > Is there a clear inconsistency with this theory? > If not, then can this theory prove Infinity?