Despite the similarity between the category-theoretic (“function-based”) and set-theoretic (“membership-based”) theories of collections discussed above, for the purpose of the structuralist these theories are crucially different. Con- sider now the language of ZF, which consists of variables of only one type, two primitive binary relation symbols ∈ and =, and no constant symbols. How are the structural properties of its objects to be characterized? As structural properties were viewed as those which did not depend on particular objects (and so in this context, those which do not depend on particular sets), one might expect that again those predicates with no parameters and no constant terms would be those which correspond tostructural properties. This view of structural properties was seen to be motivated by the two structuralist per- spectives (See Section 2.1.1). Unfortunately, a problem with this approach emerges when one considers the other characterization of structural proper- ties: those which are common to all isomorphic objects (where the objects in question are sets in the present case). On the category-theoretic approach these views were shown to coincide. Unfortunately, these two conceptions of
structural properties do not coincide in the language of set theory.
Given that set functions can be described directly in the language of set theory33, one may consider a simple case, where two sets are taken to be isomorphic exactly when they stand in bijective correspondence. Already in this case we see the two conceptions of structural property come apart. Consider (using the usual abbreviations) the predicate
Φ(x) =df ∀y(y∈x→ ∃!z(z ∈y)),
which corresponds to the property of “having only singleton members”, and
33Where an n-ary function is a set of ordered (n+ 1)-tuples satisfying the standard
take the sets A = {1} and B = {2}.34 Note that Φ involves no constant
terms and no parameters, so exemplifies the first conception of a structural property. Both A and B are themselves singletons, and so isomorphic in the simple sense of being in bijective correspondence. However, the single element of B has two members, and so Φ(B) fails while Φ(A) holds; 2 is not a singleton, and so Φ is not preserved under isomorphism.
This example illustrates a difficulty inherent to the set-theoretic frame- work that is similar to the problem of identifying the natural numbers that Benacerraf describes in [10]. In the case considered above, the language of set theory allows us to distinguish—using (predicates corresponding to) properties that do not involve particular elements—two isomorphic sets, but this distinction is irrelevant to their role35 as singletons. Qua singletons, set
A serves us just as well as set B, but the language of set theory is, in a sense, too fine grained: it allows for the formulation of predicates that are
structural in the sense of not depending upon particular objects (and so not involving constants or parameters), but which arenot structural in the sense of being common to all isomorphic sets. Further, this result can be taken to show that the characterization of structural properties as those which do not involve particular objects is inappropriate with respect to the language of set theory—the properties identified by that criterion are not structural.
The situation is no better when we take into account more complicated structures along with their associated isomorphisms, as in the case of the
ω-sequences that feature in Benacerraf’s discussion [10]. In that case, the von Neumann and Zermelo finite ordinals are equivalentas ω-sequences, but differ with respect to their set-theoretic properties. Benacerraf’s complaint is that, while the Zermelo and the von Neumann finite ordinals both stand as candidates for the titlethe natural numbers, “the accounts differ at places where there is no connection between features of the accounts and our uses
34Here taking the von Neumann definitions: 1 =
df {∅}and 2 =df {∅,{∅}}.
35Theroleof these sets can be taken to be either identified implicitly by the isomorphism
of the words in question” [10, p. 62]. The formula Φ constructed in the example above serves to illustrate one of the inconsequential differences be- tween ω-sequences that can be expressed in the language of set theory. One aspect of this difficulty is due to the language of set theory itself, as the lan- guage permits the expression of non-structural properties. In contrast, the category theoretic framework can be augmented by axioms (all expressed in the language L) that allow for the definition of a natural number object, the category-theoretic analogue of an ω-sequence. In fact, given one such natural number object, there are provably infinitely many such objects.36
The category theoretic account of an isomorphism remains as before: two objects A and B are isomorphic provided there exists an iso f : A →∼ B. Again here, it is provable that (continuing to restrict attention to predicates that do not involve constants or parameters) “All natural number objects are indiscernible in this theory. They provably have all the same properties” [61, p. 494]. Thus, taking the first account of structural property, distinct natural number objects share all structural properties when expressed in the language of category theory. Further, framed in the language of category theory, the first and second accounts of structural property coincide; those properties without constants or parameters are common to all isomorphic objects.37
To summarize, structural properties on one view may be taken to cor- respond to those predicates in which particular objects do not feature (so no constants, no parameters), or they can be taken to be those which are common to all isomorphic objects. These two views coincide in the language
36One proof of this result proceeds by taking different successor relations defined on the
same object, see [61, p. 493].
37McLarty considers the more complicated case where systems of objects and one or more
arrow are taken into consideration, and so the structural properties are not represented by predicates with a single free variable, but also include variables for any number of distinguished arrows as well. In the case of natural number objects, two distinguished arrows are admitted, one corresponding to the successor operation and one to the selection of a base-element. In [61] McLarty establishes a more complicated Structural Properties Theorem involving these natural number objects.
of category theory: all predicates in which there are no constants and no parameters correspond to structural properties—they are common to all iso- morphic objects. In the language of set theory, these two views come apart; predicates not involving constants or parameters do not all correspond to structural properties. In the language of set theory, we have recourse only to the account that identifies structural properties as those properties common to all isomorphic structures. But which are those?
In the language of category theory, there is asyntactic criterion to which we can appeal in identifying (predicates corresponding to) structural proper- ties. In the language of set theory, a predicate’s avoidance of constant terms and parameters is, as we have seen, not sufficient to guarantee its preserva- tion under isomorphism. The structuralist who appeals to the language of category theory finds that—rather than having to rely on the language of set theory in order to describe the mathematically relevant features of the ob- jects under consideration—it is the language of category theory which serves to separate the wheat from the chaff, yielding only isomorphism-invariant properties. And it is exactly the isomorphism-invariant properties that are of interest to the structuralist.
It is worth remarking here that the problem facing the structuralist who adopts a set-theoretic framework is not addressed merely by producing a criterion according to which properties of the language can be identified as being preserved under isomorphism. That structuralist faces a further question: why use a language which so readily allows for the formulation of
(predicates corresponding to) properties not preserved under isomorphism? If such properties are of no interest to the structuralist, why should they be admitted at all into the framework of a structuralist program?