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?

  • $\begingroup$ 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. $\endgroup$ – Monroe Eskew Dec 18 '18 at 11:57
  • $\begingroup$ @MonroeEskew your point is not clear? what do you want to say? $\endgroup$ – Zuhair Al-Johar Dec 18 '18 at 14:06
  • $\begingroup$ 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. $\endgroup$ – Monroe Eskew Dec 18 '18 at 14:47
  • $\begingroup$ 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. $\endgroup$ – Joel David Hamkins Dec 18 '18 at 14:48
  • $\begingroup$ @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? $\endgroup$ – Zuhair Al-Johar Dec 18 '18 at 18:20

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