From sets to classes

It seems that our the best attempt of reducing sets to classes is by taking a language with a name for each set. Indeed, note that this would be a language very different from the ones we are used for instance in set theory since it will be an uncountable language. Say ais the name for the set a, we would then have the coextensional classxˆ(x∈a)as a class surrogate for the set a. Now, it is immediate that although on some occasions such class surrogates can be found without introducing all the aforementioned constants, for instance in the case of the emptysetxˆ(∼x=x), its singletonxˆ(∼∃y(y∈x))or the powerset of the latterxˆ(∀y(y ∈x⊃ ∼∃z(z∈y))), it will clearly not be the case in general that for every set there is a coextensional property expressible in, say,_fol=,∈_.

Think for instance ofP(ω), hence the bulk of the work is done by the constants,

and so such a process can only be possible by presupposing set theory, and in so doing begging the question of the reduction.

Note that here we are already leaving aside worries that, even if we had a coextensional property with a set, this would be different since the individuation conditions of sets and classes are as we pointed out in §4.5 different. Though here one might reply that insofar as properties specified as being a member of set x, these inherit their rigidity of designation from the sets they mention and so in these particular cases extensionality does suffice for equality. Nevertheless, one could further reply to this by pointing out that these properties are somehow too close to sets to be full-blooded classes, or allowing for the terminological mismatch, to be proper classes.

Indeed, we return now to our familiar theme that classes as opposed to sets are closely tied to what is expressible given a language, while sets depend on some set operations, as well as, more generally, on the notion of well-order. This seems to be independent of our choice of language. To borrow Bernays’ point in his quote from §4.5, these exist independent of us and so, in particular, of our language. Indeed, even the notion of counting meant, for Cantor, countable by God, as Lavine remarks,15 and so is far removed from our fellow earthly mathematicians. So, as Parsons puts it such a purported reduction fails since when talking about what constitutes a set, as opposed to a class, there is an: ‘absence of any specific role for language.’16

13_{This is of course closely tied to that of whether we can reduce classes to sets for if this} was feasible the answer would be a clear no.

14_{See (Charles Parsons, 1974, ibid.)} 15_{See, (Lavine, 1994, p. 55)} 16_{See (Charles Parsons, 1974, p. 10)}

5.4. REDUCTION 101

Another problem for the reductionist seems to surface when we considering the distinctively set theoretic operation of the powerset17_{. Now, the idea here}

is that we suppose that we have obtained the class reduction of some set and we want to obtain by some analogous operation the class surrogates of all its subsets, i.e. a class containing all the subclasses. Now, here we brig into play a key different between sets and classes, for in the case of sets we now that all the members of a set are well defined sets. Indeed, as we now from Cantor’s demonstration that some collections are not sets, a collection containing an inconsistent multiplicity is itself inconsistent, and so we can say that for this operation to be carried out successfully we need to be able to quantify over all members of the set in order to collect them into the definite subsets, and in particular over these subsets since this will be the well-defined members of the new set. In fact, for any set we must be able to quantify over its members since this will be a well-defined domain. Indeed, these are crucial features of sets, as Parsons puts it: ‘ the two assumptions that get real set theory off the ground-the extensional definiteness of quantification over all subsets of a given set, and the existence of the power set-’18_.

Let’s now return to our surrogate class for the reduced set, we want to find all the subclasses, in order to construct the power class which will serve as the class reduction of the power set. However, note that it is not clear what do we mean by all the subclasses since by using the logical operators we can find new classes out of the purported totality of subclasses. Indeed, just think about diagonalisation properties or properties that quantify over all such classes. Then, to get all the classes we should really need a hierarchy of languages that build up classes on stages by performing the class operations on the classes already formed in the previous stage and so on19_{. Then, we might hope for this}

hierarchy to reach a fixpoint at some high enough ordinal stage of iteration, or if this is not to happen, just take the union over all ordinals as our final language. However, such a thing will again be not a good basis for a reduction since we would then again be presupposing the set-theoretic operations or other notions such as the sequence of transfinite ordinals.

If, on the contrary, we take all classes to be ‘all those that might be defined , independently of any specification of the means’20, as Parsons puts it, it seems that doing this would be of no help since then we would not gain much in the determinacy of the quantification over the subclasses. Indeed, what does it even mean to talk about all classes that might be formed, and how is this determinate when, under this understanding it is not even determinate the language or group of languages under use? Moreover, recall that the notion of class is closely tied to that of definable property, not just that, but to a notion of logically definable in some given language, so it seems that to have a clear understanding of the totality of classes, and of justa class for that matter, one needs a clear picture of the languages (or sequence thereof) in use. This is not the case under such understanding of the totality of classes. Indeed, here the problem does not even seem to be that we lack an understanding of the totality of classes but of even

17_{Indeed, see§4.4.}

18_{See (Charles Parsons, 1974, p. 9)}

19_{See for instance (Maddy, 1983, pp. 126-7) for something along these lines.} 20_{(Charles Parsons, 1974, p. 9)}

any class. This is being forced upon by its close ties to language and our lack of clarity with respect to this.