Forcing in Ackermann's Set Theory

Question

How would one do forcing in Ackermann's set theory? C. Alkor published an article entitled "Forcing in Ackermanns Mengenlehre" (Zeitshcr. f. math. Logik und Grundlagen d. Math., Bd. 25,8.265-286 (1979)) but the abstract, at least, is in German which I can't read. I ask the question because the purely set-theoretical part of Ackermann's set theory equals ZF and wonder whether one could use the part of Ackermann's set theory dealing with classes to construct generic extensions of models of the purely set-theoretic fragment of the theory. If one could then it would seem that Ackermann's set theory would be the proper setting for the naturalistic account of forcing.

Answer 1

If one moves to a very similar but somewhat stronger theory than Ackerman set theory, then forcing works fine.

Ackerman set theory is a version of set theory where one views the set-theoretic hierarchy as continuing far past the construction of the sets, into the construction of classes, classes of classes and so on. In the Ackerman theory, it is as though one is building the full set-theoretic $V_\alpha$ hiearchy, but then part-way through one finds a particularly robust $V_\delta$ and declares its elements to be the real "sets", with everything above $\delta$ declared "classes". (Critics would say that Ackerman's sets are only some of the sets, since his classes behave fundamentally like sets.)

As François points out in the comments, however, the Ackerman theory seems to provide less than what one may want in the realm of classes, a weakness in the theory that is addressed by its natural strengthenings to various set theories in a more ZFC-like context. Namely, the Levy theory is ZFC+$V_\delta\prec V+\delta$ is inaccessible, where $V_\delta\prec V$ is the scheme asserting $\forall x\in V_\delta\ (\varphi(x)\iff\varphi(x)^{V_\delta})$, which is expressible in the language of set theory augmented with the constant symbol $\delta$. The set $V_\delta$ here plays exactly the role of $V$ in Ackerman's theory, and so every model of the Levy scheme is a model of Ackerman set theory, if one regards the elements of $V_\delta$ as the official "sets" and the sets above $V_\delta$ as the "classes". But the Levy theory asserts more than Ackerman, because not only is the collection of sets existing as an object in the theory, but also it is an elementary substructure of the full universe. In addition, the Levy theory has a fuller treatment of classes, making them much more set-like, in that the larger universe above $\delta$, which correspond to the classes of the Ackerman theory, actually satisfy ZFC.

Meanwhile, many arguments in the literature are using forcing over models of the Levy theory (for an example, see my paper A simple maximality principle, and the Levy scheme is often arising in other work). If $\mathbb{P}$ is a set in $V_\delta$ and $G\subset\mathbb{P}$ is generic, then it is not difficult to show that $V_\delta[G]\prec V[G]$ and so the theory is preserved to the forcing extension. This argument resembles many other arguments that forcing preserves large cardinals, and one should look upon the Levy theory and Ackerman set theory itself essentially as a large cardinal assertion.

Since the idea of reflection is one of the central motivations of the Ackerman set theory, I would say that anyone tempted to use Ackerman set theory would likely prefer the Levy theory, because the form of reflection is more robust. Furthermore, the strength of the Levy theory fits right into the large cardinal hierarchy, equiconsistent with the assertion Ord is Mahlo, which is strictly weaker than a Mahlo cardinal.

If one drops the assertion that $\delta$ is inaccessible, then one gets ZFC+$V_\delta\prec V$, which is equiconsistent with ZFC. This weaker theory is strengthened by the Feferman theory, which asserts a proper class tower of such $\delta$, essentially asserting that the entire set-theoretic universe is the union of an endless transfinite elementary chain $V_{\delta_0}\prec V_{\delta_1}\prec\cdots V$. This theory is also equiconsistent with ZFC, but if one wants the $\delta$ to be inaccessible, then one gets a large cardinal strength still below a Mahlo cardinal. Because the Feferman theory provides such a robust universe concept, many authors have proposed it as a natural setting for category theory, viewing the $V_\delta$ as various universe levels.

Lastly, let me address your proposal to use the classes of Ackerman set theory to produce generic extensions of the set part. This is impossible. If $\mathbb{P}$ is a set forcing notion, then in the Ackerman theory any subcollection of $\mathbb{P}$ that is a class is already a set. So there are no new generic filters arising as classes in the Ackerman theory that are not already in $V$. The analogous situation in the Levy theory is that if $\mathbb{P}\in V_\delta\prec V$ is a notion of forcing, then every $G\subset\mathbb{P}$ in $V$ is already in $V_\delta$, and hence is not $V_\delta$-generic, unless the forcing was trivial.