# Some questions about Ackermann set theory

In a comment on this site Andreas Blass stated:

"To fit this situation into my philosophical point of view, I'd say that what Ackermann's theory calls proper classes are really certain sets. That notion has some support in the Levy-Vaught interpretation of Ackermann set theory in a conservative extension of ZF, where both the sets and the classes of Ackermann are interpreted as certain sets in the sense of ZF.

– Andreas Blass Feb 9 at 16:58 "

Does it mean that the following statements (analogs of the axioms of pairing, union and powerset respectively) are consistent with Ackermann set theory:

1) For any two classes $X, Y$ there exists the class $Z$ which contains just $X$ and $Y$.

2) For any class $X$ there exists the class, whose members are just the members of the members of $X$;

3) For any class $X$ there exists the class whose members are just all the subclasses of $X$.

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@wmitt: Thanks for the answer. Could you provide some references to theorems supporting your answer? Does it follow from your answer that there exists a conservative extension $AZ$ of Ackermann set theory which ($AZ$) is also a conservative extension of $ZF$? –  Victor Makarov Apr 20 '12 at 15:37