# Can reflection in $V$ and reflection out of $V$ principles be used in the same theory?

I've noticed that most of the work in Ackermann set theory is primarily concerned with constructing sets in $$V$$, the rest of the classes are just excess material, carrying no comprehension over them. There is a try of Muller in which he strengthen the class existence principle of Ackermann into Separation over classes, the resultant theory is $$A$$, and adding Regularity $$R$$, and Choice $$C$$, he gets into $$ARC$$, a theory claimed [see here] to serve as a foundation of both category and set theory, and thus for most of mathematics.

This gave me the idea of reflecting-out of $$V$$ principle, since Ackermann's set theory can be interpeted in systems using reflection [see here] , so if to any of the two systems appearing in that posting (with reflection in them re-named as reflection in $$V$$), we add the following principle:

Reflection out of $$V$$ schema: if $$\varphi$$ is a sentence in $$FOL(=,\in)$$, i.e. doesn't use the symbol $$V$$, and $$\varphi^V$$ is the bounded by $$V$$ sentence of $$\varphi$$, i.e. the sentence obtained by merely bounding every quantifier in $$\varphi$$ by $$V$$, then: $$\varphi^V \to \varphi$$, is an axiom.

In other words we are reversing the reflection process, so we are concluding things about classes in general by reflecting from the inside of $$V$$ to outside it. By that, all set axioms (i.e. sentences in the language of set theory that are satisfied in $$V$$), would generalize over all classes. This way we easily get to interpret Muller's theory.

Question: is there an obvious inconsistency with a theory that both uses reflection in $$V$$ and reflection out of $$V$$ principles?

• What about a schema with a constant symbol $c$ asserting that $c$ is some $V_\alpha$, and $V_\alpha$ is elementary in $V$? This is a mild theory. – Monroe Eskew Dec 18 '18 at 11:57
• @MonroeEskew your point is not clear? what do you want to say? – Zuhair Al-Johar Dec 18 '18 at 14:06
• Fix some $V_\alpha$ satisfying ZFC and pretend it is $V$ and the higher-rank sets are hyperclasses. Possibly $V_\alpha \prec V$. In such a situation, we have reflection out of $V$. Also everything true in $V$ reflects to $V_\alpha$. We can also have this on a club of $\beta$, so that also things can reflect below $\alpha$ simultaenously. – Monroe Eskew Dec 18 '18 at 14:47
• See mathoverflow.net/a/103779/1946 for more discussion. Also, this theory arises in many other questions on MO. See mathoverflow.net/search?q=user%3A1946+Feferman+theory. – Joel David Hamkins Dec 18 '18 at 14:48
• @MonroeEskew do your comments apply to adding the reflection out schema on top of the referred theory in the post that has the limitation of size axiom? this has a mahlo cardinal as its model, (see link in the head post), so is this also mild in your opinion? – Zuhair Al-Johar Dec 18 '18 at 18:20