This posting comes as a possible salvage to this earlier presented reflective theory which was proved inconsistent. Attesting the particulars of the underlying language in reflection axioms.

Working in *mono-sorted* first order logic with equality $``="$ and membership $``\in"$:

*Define:*$set(x) \equiv_{df} \exists y \, (x \in y)$Axiomatize:

**Extensionality:**$( a \subseteq b \land b \subseteq a \to a=b)$**Separation:**$(set(a) \to \exists \ set \ x : \forall y \, (y \in x \leftrightarrow y \in a \land \phi ))$**Reflection:**$ (\varphi \to \exists \ set \ x : \text { trs}(x) \land \varphi^x)$

where formulas $\phi, \varphi$ do not use $``x"$; $\varphi$ do not use $``="$; $\varphi^x$ is obtained from $\varphi$ by merely bounding its quantifiers by $``\subseteq x"$ if the quantified variable appears only on the right of symbol $\in$, and by $``\in x"$ whenever it appears on the left. $``\text { trs}" $ stands for *is transitive*.

Can this syntactic restriction on $\varphi^x$ manage to escape inconsistency?

I conjecture that this reflection scheme would be equivalent to second order Bernays reflection schema (page 21) over the rest of axioms of this theory.