Ackermann

In this section we deal with the theory _a, Ackermann’s set theory, as well as others introduced in§2.6. Although in_awe also notice the problem that both sets and classes are individuated extensionally, we readily notice an advantage of this systme with respect to the previous theories of sets and classes just considered. Namely, that here, although we will also have a set predicate, this will not be definable. Recall that this was justified by Ackermann taking seriously the idea that the extension of the set predicate is never completed and varies with time and so we should not at any given point be allowed to fix its extension. We also saw this can be modelled using intutionistic Kripke frames. In particular it will not be definable in terms of membership as in_ml,_nbg, or

mkand so we seem to have as desired a more genuine distinction between sets and proper classes. Indeed, as we also proved in §2.6 there are proper classes that contain other proper classes and so this is something that we welcome.

Notwithstanding the previous remarks, we see that this theory still has lim- itations that we cannot accept. For instance, consider the class comprehension principle, as we saw in the case of _nbg it prevents Russell’s paradox by speci- fying that the elements of this new class will just be sets. But of course, we do not endorse the restriction with regards the classes introduced that these should only be introduced through properties whose instances are sets. Our compre- hension principle must not discriminate a priori between properties satisfied by sets and those by classes when it comes to deciding which entities are included in the new class. Indeed, although this principle would guarantee a class of all sets, it would not guarantee a universal class, i.e. a class including all sets and classes, as we require. Related to this point we see that since we cannot rely on this principle of comprehension to introduce classes containing other classes, we cannot prove the fact that some class contains some other class constructibly, by for instance considering the universal class. We must do this by appealing to a contradiction via the definability of sets. So in sum even if this theory allows a class to contain another class it does so in too an uninformative of a way to be considered satisfactory.

There are also serious worries here with the notion of set Ackermann is employing. Indeed, we see that as opposed to the Cantorian notion of set we espouse, for Ackermann the axiom guaranteeing the existence of sets is com- pletely devoid of any flavour of limitation of size. Even leaving this aside, as we noted in §2.6, it seems arbitrary that since for Ackermann formulas with class parameters are ill formed sentences, since recall that since the notion of set is being always sharpened the range of these would not be stable enough, but allows quantification over all of them. Given our adoption of the all in one principle the domain must be a class and so ill formed as well according to him. In any case, his set comprehension schema seems much closer to a reflection

principle. This makes sense when we consider his idea of the notion of set in continuous development, and so embodied in the fact that if at a given point some class has only sets as members, in the next step of the hierarchy it will be a itself coextensional with a set. Indeed, if the reader forgives the excursion into

zfc, inVω, there is an infinite class of sets, and thisωis a set inVω+1. It seems an interesting question whether reflection principles fit in with the Cantorian idea of set, lacking a more exhaustive analysis of the issue, it seems however that closer to the spirit of Cantor’s ideas would be the notion of transfer prin- ciples such as those of Friedman, as seen in§5.2.2. Indeed, reflection principles approach the hierarchy from above.This we see in the proof of the axiom of infinity of _a through an appeal to the universe of sets. This would correspond to Cantor’s transfinite and it seems at least dubious that appealing to such a, for him, unknowable entity to prove some facts about sets would be felicitous in this framework. Be this as it may, the notion of allowing a comprehension principle not alluding to size to be governing the existence of sets seems to us too close to the notion of class to find it satisfactory. In sum, in this theory even though the classes are closer to our idea of class, they seem rather obscure entities with regards to membership matters. It also seems that the notion of set is too close to that of class.

We now look briefly into Reinhardt’s theory s. Note that one can interpret Reinhardt’s talk of imagined entities as what we understand as classes. Even if not a technical problem this can already prompt a more ontological protest from us since we do not take classes as less real than sets as this denomination would suggest but rather as equally real albeit different. Remember, incidentally, that Ackermann takes the real subject of set theory to be sets and not classes, indeed for us a satisfactory theory ought to take into consideration both entities and do so seriously. In any case, the_sschema characeristic of this theory is a reflection principle similar to the comprehension for sets in_a, although stronger since here the objets satisfying the property are not required to be sets like in Ackermann’s case, and so our worries doubting this is a principle respecting Cantor’s notion of set limited by size are still a negative feature carried into this system. Also note that the class comprehension schema of _sis less adequate than that of _a since it embodies the idea of limitation of size, note in fact that the universal class of sets cannot be introduced through this principle, and so is added to the language as a constant. Hence it seems that this theory is actually less acceptable thana.

Finally, we also mentioned the stronger theory of Powell, _p. This is an interesting theory in the sense that it adds a predication symbol,3, informing us that a class is predicated of a set. However, the fact that we have noted already in multiple occasions that sets and classes are not well-distingushed entities resurfaces, for here classes are just taken to be subsets of the universe of sets for which the language provides the constantV. Hence, although when talking about sets it is the same to say that a set belongs to a class or the class is predicated of the class. For classes no such axiom is present. Of course, we reject this since we seem to be taking classes as too close to properties since even if a class is predicated of the property giving rise to some class we want to say that it actually belongs to the given class not that it is predicated of it, this seems to be some kind of categorical mistake. In short, here sets seem to be

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taken more seriously than classes in ontological terms since only the former are allowed to belong to classes, this we do not take as a good principle. Other usual points of friction here include extensionality for classes as well as restrictions to the class comprehension of the theory that we do not accept, such as that classes can only appear to the left of predication.