Sets, Classes, and Pluralities

Sets are the paradigm abstract objects and the working ontology of modern mathematics. They are also the entities most often reached for when a nominalist needs a replacement: properties become classes of their instances, propositions become sets of possible worlds, structures become set-theoretic constructions. This makes their own status a load-bearing question — a reduction to sets is only an economy if sets are cheaper than what they replace.

This page takes up three questions about them: what conception of set we are working with, whether proper classes are objects at all, and whether quantifying over many things really requires a single thing that collects them.

Two conceptions of set

The logical conception treats a set as the extension of a concept: for any predicate, the set of things satisfying it. This is Frege's route, and it makes sets the natural correlates of properties — the connection that class nominalism exploits. It is also inconsistent. Unrestricted comprehension yields Russell's paradox: let $R=\{x:x\notin x\}$, and $R\in R$ if and only if $R\notin R$.

The iterative conception replaces it. Sets are built in stages: begin with nothing (or with urelements), and at each stage form all sets of what is already available, continuing through the ordinals. The cumulative hierarchy

$$V_0=\varnothing ,V_{\alpha +1}=\mathcal{P}(V_{\alpha }),V_{\lambda }=\bigcup _{\alpha <\lambda }{V}_{\alpha }$$

contains every set at some level, and no set is ever a member of itself, because a set is always formed after its members. The paradox does not arise: there is no stage at which $R$ could be formed. This is the conception the ZFC axioms codify, and its ontological significance is that it makes sets ontologically posterior to their members. A set depends on its members, and not conversely — a paradigm of the dependence axis.

The philosophical moral is worth stating plainly. The paradoxes did not show that there are no sets; they showed that the logical conception, on which sets track predicates, was wrong. What survives is a conception on which sets are constructed objects with a definite order of formation — much less useful for the nominalist who wanted a set for every predicate.

Proper classes

Some collections are too big to be sets: the collection of all sets, of all ordinals, of all things self-identical. Treating them as sets reproduces the paradoxes. Three responses, with different ontological content:

• ZFC: there are no such objects. "The class of all sets" is a façon de parler, eliminable in favour of formulas; talk of proper classes is talk about predicates, not about entities.
• NBG / MK: proper classes are genuine objects, but of a second sort — they can have members but cannot themselves be members. NBG is conservative over ZFC; MK is not, and proves Con(ZFC).
• Indefinite extensibility (Dummett, after Russell): the trouble is not size but that any determinate totality of sets can be diagonalised to produce a further set. The concept set is one whose extension can never be closed off, so "all sets" never refers to a fixed domain.

The last has consequences beyond set theory. If some concepts are indefinitely extensible, then absolutely unrestricted quantification may be impossible — there may be no such thing as quantifying over absolutely everything, only over progressively larger domains. That threatens ontology at the root, since the question "what is there?" was supposed to be asked unrestrictedly, and the criterion of commitment assumes a determinate domain for the variables to range over. It also bears on whether the quantifier is univocal.

Plural quantification

A standing assumption of the Quinean framework is that to quantify over many things is to quantify over one thing that collects them. Boolos argued that this is false, and the point has direct consequences for ontological commitment.

Consider the Geach–Kaplan sentence:

Some critics admire only one another.

It has no first-order paraphrase in terms of the critics alone; the natural rendering quantifies over a set of critics. But the sentence appears to say nothing about sets — only about critics, plurally. Boolos' proposal is that plural quantification is a primitive logical device: "there are some things such that…" ($\exists xx$), with no commitment to a further entity that is the many.

Two consequences matter here.

It loosens the tie between quantification and commitment. If plural quantification is legitimate logic, then a theory can quantify over many things without being committed to a collection of them, and Quine's criterion needs restating: commitment attaches to what the singular variables range over. The nominalist gains a resource — many things without a set of them — and the ostrich gains a way to say quite a lot while committing to little.

It rehabilitates second-order logic. Boolos read monadic second-order quantification plurally, which answers Quine's charge that it is "set theory in sheep's clothing": $\exists XXa$ says there are some things among which $a$ is, not that there is a set or property. Whether the plural reading is genuinely innocent, or covertly quantifies over collections, is the live dispute; the sceptic notes that plural logic has a semantics stated in set theory, which the defender answers by denying that the semantics is what confers meaning.

Sets versus sums

Sets are not the only way to get one thing from many. Mereological sums are the other, and the contrast is instructive because it shows how much structure set membership carries.

Two features stand out. First, the singleton: $\{a\}\ne a$, and there is an infinite ascending tower $a,\{a\},\{\{a\}\},\dots$ of distinct objects generated from one. This is where the set-theoretic hierarchy comes from and where its ontological extravagance lies. Lewis, in Parts of Classes, showed that granted the singleton relation the rest of set theory is mereology — a class is the sum of the singletons of its members — so the whole mystery of sets is concentrated in the single step from an object to its unit set, which he confessed he did not understand.

Second, transitivity. Parthood is transitive, membership is not: a molecule of a tomato is part of the sum of all tomatoes, but not a member of the set of tomatoes. This is precisely what makes sets fit for structure and sums unfit — and why mereological nominalism fails where class nominalism at least gets started.

Lewis's further claim, that mereology is ontologically innocent — that in accepting $a$ and $b$ you have thereby accepted $a+b$, since the sum is nothing over and above them — is the basis of the argument for unrestricted composition.

Where sets sit

Sets sit at the junction of the inventory and category questions, and their peculiarity is that they are so often used as small change in other ontological transactions. Properties reduced to classes, propositions to sets of worlds, structures to set-theoretic models: each reduction is an economy only if sets are already paid for. Since the iterative hierarchy is enormous — vastly larger than any inventory of universals a sparse realist would countenance — the accounting is not obviously in the reducer's favour. This is the point made against class nominalism on the nominalism page.

Plural quantification is the more disruptive item here, because it touches the section's method rather than its contents. If quantifying over many things is not quantifying over one, then a criterion of commitment that reads ontology off the variables must say which variables, and the answer is not supplied by the criterion itself.