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?


Yes, there is a clear inconsistency, take the formula:

$\exists A.\ (\forall x.\ (x\in A \leftrightarrow A=\emptyset))$.

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 $A\neq \emptyset$.

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