# Can we have cyclic generalized positive comprehension?

In positive set theory, the axiom scheme of generalized positive comprehension in $$GPK^+_\infty$$ [of Olivier Esser] is stated in a manner as to forbid the symbol of the asserted set to occur in the defining formula.

To quote from the above source:

Positive Comprehension: For any positive formula $$\phi$$ in which $$A$$ does not appear, $$(\exists A.(\forall x.x \in A \leftrightarrow \phi)).$$

Where a positive formula is defined as:

Deﬁnition: Let the class of (bounded) positive formulas be the smallest class containing the formula $$x \neq x$$ (useful because uniformly false), all atomic formulas, and closed under conjunction, disjunction, bounded universal quantiﬁcation $$(\forall x \in A.\phi)$$ and existential quantiﬁcation $$(∃x ∈ A.φ)$$.

Question 1: Is there a clear inconsistency involved with permitting symbol $$A$$ to occur in $$\phi$$, in the statement of positive comprehension?

Question 2: Suppose that there is no clear inconsistency, then would that allowance results in increment in consistency strength over the original system?

$$\exists A.\ (\forall x.\ (x\in A \leftrightarrow A=\emptyset))$$.
Clearly, such a set $$A$$ cannot exist: if $$A\neq \emptyset$$, then any $$x\in A$$ would give a contradiction. If $$A=\emptyset$$, then this formula would imply that any $$x$$ satisfies $$x\in A$$, but then $$A\neq \emptyset$$.