# 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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## 1 Answer

The answer to your question is "Yes."

In trying to understand why Ackermann nonetheless wanted to distinguish proper classes such as V from sets - where V is the proper class of all sets, taken here to be a natural model of ZFC - it is necessary to take seriously Ackermann's claims about the universe V being continually "under construction," so to speak, or always in the process of being "built" (see Penelope Maddy's article "Proper Classes" (Journal of Symbolic Logic, 1983), p. 122, on this point). At some particular "time" (or "step") t in this construction process, there may exist (e.g.) classes larger than V ("superclasses") that have been obtained by iterating the power-set operation on V denumerably many times. Consider the totality T of such superclasses existing at t; i.e., T (at t) is the limit of a denumerable sequence of iterated power-set operations on V. Since T (at t) is clearly not a natural model of ZFC, it cannot be regarded as a new "universe" that supersedes or replaces V. Hence, those collections which (at t) are members of T but not of V are not members of a suitable universe or natural model; and so they are to be distinguished from the members of V, a distinction expressed by calling them "proper classes" (and calling the members of V "sets").

One glaring problem with this account is that it's unclear what we're supposed to take as the time t. Why, in particular, should we suppose that t is such that the power set operation on V has been iterated denumerably many times (or even some nonzero number of times)? Without any constraints on t, there's simply no way to know what proper classes we're supposed to regard as existing; and, in fact, there are no theoretical constraints on what t's value is. Hence, there's no basis for saying anything definite about proper classes at all; and so it seems best, as Andreas Blass said, to treat Ackermann's "proper classes" simply as sets.

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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
There's F.A. Muller's "Sets, Classes and Categories," Brit. Jrnl for the Phil. of Sci. 52 (2001), 539-73. Muller adds a Class Separation Schema ("ClsSep") to Ackermann's theory ("A"), and this yields the desired analogs of Pairing, Union and Powerset (565). Contrary to Muller (564), I believe his theory is a conservative extension of A (and of ZF), since (i) ClsSep entails Ackermann's class existence schema ClsEx, and (ii) ClsEx is just the restriction of ClsSep to classes that contain only sets. Cf. A. Blass's review of Muller at MathSciNet (MR1851712), 3rd paragraph from the end. –  wmitt Apr 20 '12 at 21:49
Btw, note that Muller combines A with Regularity for sets, which gives a conservative extension of A (see Levy & Vaught). Also, it's clear from Reinhardt that any conservative extension of A is, ipso facto, a conservative extension of ZF. (The references for Levy-Vaught and Reinhardt are in the comment by Blass linked to above.) Finally, Muller tries to explicate Ackermann's set/proper class distinction by formalizing the idea of unsharpness; but he only formalizes "X is unsharply distinguished from Y," and not "X is unsharp"- and it's the latter that needs formalization. –  wmitt Apr 21 '12 at 15:27
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