From classes to sets

After giving some reasons for rejecting the possibility of reducing sets to classes in the previous section we now occupy ourselves with the converse notion, namely, can we reduce classes to sets? The idea of the reductionist here seems to be to understand statements regarding all sets as quantifying over some suit- able large cardinal but such that it is, analogous to Russell’s notion of typical ambiguity, not clear over which such cardinal this quantification is taking place. This is what Parsons calls a vague understanding of the quantifiers21_{. Given}

this understanding of the quantifiers, even when discussing properties that give rise to classes with an extension too big to be sets such as [λx(x= x)], once we make the statement more precise, ranging over some set, we can see that even these classes as a higher layer of sets. Indeed, suppose the statement is made precise by ranging overVκ withκ some inaccessible strong enough that the speaker is not aware of its existence.22 23 Then, the class will be a set once we get to an interpretation of the quantifier when this ranges over Vκ+1, since the extension ofxˆ(x=x)is the domain of discourse, which was in that caseκ, this being a set inVκ+1. Now it is then very clear how this would reduce classes to sets, for what we take as classes are just sets we have not yet grasped.

There are several ways to resist this reduction. First, we could say that given our conception of class, the reduction rests in the concept we have of set, and so it is not sure that this could be obtained in general, but that only in certain theories of sets. Not, in particular, in a well-founded theory. Indeed, suppose we consider again [λx(x =x)], since clearly [λx(x= x)] = [λx(x= x)], then

ˆ

x(x=x)∈xˆ(x=x), and suppose further that we have done the reduction of this class toκ, then we would have thatκ∈κ, which is simply false in_zfc. That the reduction cannot be carried out in the most common theory of sets, which seems after all what Parsons tries to do since he talks about the von Neumann hierarchy as the universes of quantification, is not to say that this cannot be carried out at all, specially since we do not take sets to be necessarily well- founded, but perhaps indicates that classes of a more traditional theory such as

nbg where self-membership of classes is disallowed would be less resistant to a

purported reduction. Hence, we should pay special attention to make sure that reductionist arguments take into account, in order to be successful, the specifics of our notion of class.

Note also that the reduction that Parsons is suggesting would be perhaps not what we would at the outset thought of a successful reduction. Indeed, suppose we have the classxˆ(x6=x), then we would look at∅as the adequate reduction of

21_{See (Charles Parsons, 1974, p. 10) and (Charles Parsons, 1983, p. 521).}

22_{Though, one must be, according to Parsons, careful since this precisification does not} exactly capture the speaker’s intention, for we seem to be sharpening the domaintoomuch, see (Charles Parsons, 1983, p. 523).

23_{Moreover, the idea behind this proposal should remind the reader of the intuitionistic} semantics for set theory explained in§2.

5.4. REDUCTION 103

such class. Here we use a definite description since one can take the reductionist to be saying that to each class there corresponds one and only one set. However consider again our class xˆ(x=x), this will be reduced to someκ. Now, what Parsons says is that which particularκthis turns out to be is ambiguous. But it would be some definite κor another. However after realising this we again want to account for our universal quantification, accordingly, we will be now quantifying over some other inaccessible cardinal κ0_{, such that κ}0 _{> κ, now}

then this class will be reduced to some κ0_{. The idea here might be that the}

class reduced toκ0 is a new class and so it makes sense that it is assigned to a different set, since one class includes more sets than the other now that our understanding of the notion of sets has grown to a new inaccessible. However, note that, with our view of classes, this reply is not tenable since we take classes to be intensional entities closely tied to properties. Even if the extension of the class is different when taking the universe to beVκ than when it isVκ0, this is

the same class since the property defining it[λx(x= x)] is the same in both cases. And so repeating this thought process we see that this class has been reduced to a infinite number of sets and all of them different. It is hard to see then how a picture like this can be seen as a satisfactory reduction since, it is not only that we do not know exactly to which set the class corresponds, but that we do not know to which infinite family of sets we have reduced the class. It is not clear how such a picture can give us a practical way to explain away our talk of classes. Indeed, if asked to reinterpret talk about the empty class I can just talk of the properties of the empty set, if asked about the universal class in what seems to be Parsons understanding I will just refer to some cardinal albeit not knowing which, but in our case we have lost also the definiteness of reference.

One might think that we are exaggerating the problems of such a reduction by drawing our attention to the case of structuralists about numbers24_{, (at least}

when we are not talking aboutante-remstructuralism) the idea here is that we reduce numbers to certain positions on isomorphic structures, hence as in the case we are interested with here, what seems like one object in our discourse has been reduced to a plurality of entities. However, note that the situation here looks more problematic since what for the structuralist brings these entities in the reduction together is to be certain positions they share on some isomor- phic structure. So one might want to know what is the analogous thing these large cardinals share? Certainly nothing like that, for the only thing they have in common is that at some point the speaker had an understanding of set such that this could be formalised in a truth preserving way by a model of_zfcof that cardinality,Vκ. So unlike the structuralist reduction this seems a sui-generis re- duction, relativised and particular to each of the mathematical subjects, and seemingly not capable of objective verification as is straightforward in the case of structuralists by means of finding an isomorphic map. In short, in the struc- turalist case there seems to be a clear tie between the entities that constitute the number, which seems to be lacking for the multiple sets corresponding to the class.

Now, the previous remarks were an attempt to undermine the reductionist

picture by focusing on features of our conception of class, perhaps it is more direct to oppose Parsons idea that one is allowed to take talk of universal quan- tification over sets to be talk over some Vκ for a suitably large κ. Indeed, one might resist the idea that preservation of truth is enough to preserve the intended meaning of someone’s utterances and assertions. As is familiar to read- ers acquainted with the debate regarding model-theoretic or Quinean arguments undermining the determinacy of reference, such positions come with a cost.25. Indeed, that this is the case is plain in the mathematical camp by thinking about the opposition to nonstandard models of arithmetic, these models pre- serve truth but might seem to be missing something significant nevertheless. As Boolos puts it in our case:

reinterpreting what they say in such a way that it is not about all sets is changing the meaning of what was said if not the truth value; (. . . ) [this] would be to misrepresent what they said. (Boolos, 1998, p. 31)

Hence, that Parsons reductionist account preserves truth is not a sufficient condition to take it as satisfactorily. In fact it seems to me that Parsons is con- flating the distinction between not knowing that some set, like a large cardinal, is actually a set from the fact that this set is part of the universe. If my inten- tion is to quantify over all sets, I will quantify over the inaccessible even though at some point on time I am ignorant of this fact. Upon becoming persuaded this, only my epistemic state with regards to my domain of quantification will change, not the domain itself. Quoting Boolos again:

But why should we believe that this account of the matter is the correct one rather than the simpler one: that at to and t he quantified over the same sets and at ti believed something about those sets (viz., that one of them was an inaccessible) that he did not believe at to? (Boolos, 1998, p. 33)

Be that as it may, as Boolos also mentions26 _{this relativisation of reference}

can be seen as going further than what Parsons seems prepared to accept. In- deed, he only talks of interpretations ranging overVκ, forκinacessible, indeed assuming the consistency ofzfc, according to the downward Löwenheim-Skolem theorem, there will be a countable model of this theory. Now, Parsons takes the universe to be some inaccessible cardinal but if this ambiguity is taken seriously and the only requirement is that the interpretation makes the statements true how could we rule out a model where the universe is a countable set? The dilemma seems to be that either we accept quantification over all sets, which seems to be also supported by set-theoretical practice. Indeed, there is no warn- ing of ambiguity when a textbook presents a theorem as holding for all sets. Or we cannot rule out that the model we are quantifying over is a set we seem to be not intending to quantify over such as a countable set. This however seems to much ambiguity to accept happily, for one thing is that one does not grasp the theory of inaccessible cardinals, and so what he takes to be all sets is just one of those, but another is to make the speaker accept that they might be wrong in their understanding of Cantor’s theorem and that actually they do not quantify

25_{The interested reader on this more general debate might consult (Williams, 2008) or} (Mcgee, 2005).