FEDERACIÓN DE ASOCIACIONES DE MUJERES SEPARADAS Y

into the language of_zfc. Namely, to include a function symbol in the language and add an axiom schema that reflects the fact that this map is an elementary embedding, quoting Jech:

As the statement “there exists an elementary embedding of V ” is not expressible in the language of set theory, the theorem needs to be understood as a theorem in the following modification of zfc: The language has, in addition to∈, a function symbolj, the axioms include Separation and Replacement Axioms for formulas that contain the symbolj, and axioms that state thatjis an elementary embedding ofV (Jech, 2003, p. 290)

It is clear that, even if in this case the classes can be disposed of, this process seems rather artificial, for they are, through model theory, valuable in the development of principles regarding these cardinals, or as Uzquiano puts it although

sometimes eliminable, but which nevertheless seem heuristically indispensable (. . . ) In- deed, set theorists often begin to work within an informal theory of sets and classes, and then search for technical formulations within either ZFC or some schematic extension thereof. (Uzquiano, 2012, pp. 70,72)

Not only that, but these theories seem more cumbersome to use than a theory of classes where one can use the second order machinery available to quantify at will over functions such as the ones under discussion here. So one could press the point further of why one should not take classes seriously given their important role in the development of the theory, but be content with what seems a much more artificial andad hocformulation which seems to be preferable just because it does away with the classes.

In view of all these examples showing how mathematicians employ resources of second order logic in their daily business, one could concede that point while still insist that we needn’t take them as committed to the existence of classes. Or press the point further by claiming that they are not really talking about classes when they make statements about all ordinals and the such, i.e. the use of classes must, or at least could, be paraphrased away. More precisely, the use of second order resources could be explained in some way that does not ontologically commit us to these entities distinct from sets. This will be explored in§5.4−5, below by means of reduction of classes to sets or the use of plurals. Nevertheless, our aim will be to convince the reader of the plausibility of accepting these ontological commitments. Now, one reason why one might not want to take classes as genuine entities that must be taken seriously is that, as opposed to sets, they are not well-motivated entities. They would be just some patches that we employ to fix a flawed theory, i.e. that classes aread hoc, this is the focus of the next section.

5.3

Ad hocness

We take as a representative example of this argument against the use of classes the remarks Jonathan Lear makes in the introduction to his articleSets and Semantics. Lear points out that modern theories of collections usually

employ both the notion of set and of proper class, in order: ‘to reconcile the intuition that any well-determined objects can be collected together and the classical interpretation of the universal quantifier.’9. Indeed, the idea is that the universal quantifier is taken to range over all sets, and that any collection of well-determined sets can be collected into another set. Now, even if dismiss- ing the last query by pointing out how Cantor’s principle of limitation of size acknowledges the fact that some collections of well-formed entities such as the ordinal or cardinal numbers are too big to do so. The fact seen in §3 that we do accept the fact that the universal quantifier ranges over all sets, together with our endorsement of the All in one principle (AiO), does indeed prompt us to affirm that there is some object collecting all sets. However, this respect for the limitation of size principle does inform us that this will not be a set. This last idea is something that most theories of collections agree with and, as Lear points out, these usually take the position, either implicit such as_zfc, or explicitly, such as_nbg, that this object is a proper class.

Before proceeding note that what Lear is talking about is a class of all sets, strictly speaking all we are sure to affirm is that there is a class that contains all sets, namely [λxx = x]. But not only this, since it will also contain all classes, hence the existence of this class, given that we take the logical notion of class seriously, will be contingent upon having a language that is allowed to express the notion of being a set, for instance via a predicate as inzfcora, or a constant such as in Maddy’s theory.

Lear thinks that this use of classes in order to account for the problem just mentioned is ‘unacceptable’. Firstly, he says because:

classically interpreted quantifier must indeed range over all sets. But it is not necessary for the quantifier to range over an object that contains all sets. (Lear, 1977, p. 87)

Indeed, this is just expressing his rejection of the AiO principle that we defended from Cartwright’s remarks in §3, and so need not detain us here. Next, Lear seems to complain that the use of classes does not clarify the issue at stake since they seem an ad hoc object, given that the restrictions imposed on them in the usual theories of collections seem unmotivated, as he puts it:

the standard restrictions imposed—e.g., that proper classes can only have sets as members, that one cannot perform set-theoretic operations on them—appear arbitrary.(Lear, 1977, ibid.)

Indeed, I agree with Lear that these usual restrictions on classes are deeply unsatisfactory, however, I think that with the discussion about the Russellian notion of class in §4, we can respond by saying that we do have a clear un- derstanding of the notion of class or at least ofour notion of class and that it is precisely our task to elucidate which of these conditions usually imposed on classes are acceptable or not. Indeed, it is precisely the close tie between prop- erties and these entities on the logical notion of class which is historically very different from that of set that allows us to face any criticism ofad-hocness. For instance, we are ready to say that we must reject that classes can have only sets as members, as our remarks above about the universal class show. Of course,