If we define Transpassing in the following manner:
$\phi$ is Transpassing $\iff$ $\exists x,z (x=\{y \in V | \phi(y)\} \wedge \phi(z) \wedge x \subset TC(z))$
Where clearly $\phi$ is a predicate symbol, $TC(x)$ stands for "the transitive closure class of $x$" defined in a customary manner as the minimal transitive superclass of $x$, and transitive class is defined as a class that has all of its elements being subsets of it.
Now define Reflective as:
$\phi$ is reflective $\iff $ $\forall x (\phi(x) \to x \in V) $
Now $V$ is a primitive constant symbol denoting the class of all sets, as in Ackermann set theory.
Now let's work in a first order set theory in the language of Ackermann's set theory, that has the following axioms:
- Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x=y$
Class comprehension scheme: if $\phi(y)$ is a formula in which $x$ is not free, then all closures of: $$ \phi \text{ is reflective} \to \exists x : x=\{y|\phi(y)\}$$; are axioms.
Transitivity: $\forall x \in V(x \subset V)$
Acyclicity: $\forall x \in V: \neg x \in TC(x)$
Acyclic set construction scheme: if $\phi(y) $ is a formula in which $V$ does not occur, and where $y,z_1,..,z_n$ are all of its free variables, then: $$\forall z_1,..,z_n \in V \\ \neg (\phi \text{ is transpassing })\to \exists x \in V :x=\{y| \phi(y)\} $$ is an axiom.
Now this theory would clearly prove all axioms of Ackermann's set theory (except the full consequence of Regularity), including the second completeness axiom for $V$. So the above acyclic set construction principle is indeed stronger than the reflection scheme of Ackermann, to write the later, it is:
Ackermann's reflection scheme for set construction: if $\phi(y)$ is a formula in which $V$ does not occur, and where $y,z_1,..,z_n$ are all of its free variables, then: $$\forall z_1,..,z_n \in V \\ \phi \text{ is reflective }\to \exists x \in V : x=\{y| \phi(y)\} $$ is an axiom.
It is easy to prove that in this theory every reflective predicate (given the conditions of not using the symbol $V$, and parameters standing just for sets) is non-transpassing. But does that hold in the opposite direction?
My question is: Does Ackermann's set theory prove all axioms of this theory? In other words, can we prove in Ackermann's set theory that every non-transpassing predicate (with above qualifications) is a reflective predicate?
The idea is that the theory presented here is based intuitively on a separate notion from that of Ackermann's, that of defining classes of acyclically constructed sets, though it overlaps with Ackermann's in it being essentially a class theory with set-hood taken to be primitive, and it shares with it all of its first four axioms. However, the notion of acyclicity is intuitively different from that of reflection; here all axioms pivot around the theme of acyclic construction while in Ackermann's two axioms seems to be deliberately fixed to suit proving axioms of union and power, even Regularity doesn't seem to be necessary for Ackermann's. However, should the answer to my question be in the positive, then the above theory would only be a rather long reformulation of Ackermann's set theory though reflecting a more unified theme of axiomatization?
