Two notions of sets

Sets are considered fundamental mathematical objects, because in principle all other mathematical objects can be defined in terms of sets (can be reduced to sets). However, the notion of a set can be given at least two unequivalent interpretations. In one interpretation, the set of physical objects X is a physical complex whose spatiotemporal parts are all objects X. Sets of that sort are called mereological (or collective). Mereological sets have the following characteristic properties. First, the mereological set consisting of one object is identical with this object. Second, there is no mereological empty set (for instance, the collection of all centaurs does not exist). Third, two mereological sets built out of numerically different objects can nevertheless be identical. For instance, the mereological set of two hydrogen atoms is identical with the mereological set of two protons and two electrons constituting those atoms. This example also illustrates the fact that the relation of belonging to a mereological set is transitive. Clearly, this follows from the fact that the relation of being a member of a mereological set is identical with the part-whole relation, and the latter is transitive (if x is a part of y and y is a part of z, then x is a part of z).

The second interpretation of sets is called distributive (or set-theoretical). Distributive sets are analogous to linguistic concepts. The distributive set of all people has people as its only elements. No proper part of a person (such as a hand or a leg) belongs to this set, since proper body parts are not humans. As concepts differ from the objects subsumed under them, the set consisting of only one object is numerically different from this object. There is an empty set (a set with no elements), since there are empty concepts (such as the concept of a unicorn). Actually, as we will soon see, it can be proven that there is exactly one empty set. If you have two sets which have different numbers of elements, you can be sure that these sets are different. Hence the set of two hydrogen atoms is distinct from the set of their two protons and two electrons. Distributive sets satisfy the principle of extensionality: two sets are identical if and only if they have exactly the same elements. (Actually, mereological sets satisfy this principle too: two mereological sets are identical iff they have the same parts. But it is still possible to describe one and the same mereological set as consisting of numerically different objects, as in the example with two hydrogen atoms). From the condition of extensionality it follows that there is exactly one empty set. The relation of membership is not transitive in the case of distributive sets: if x is a member of y and y is a member of z, x does not have to be a member of z (although it may). Example: 1 є {1} and {1} є {{1}, {2}}, but it’s not the case that 1 є {{1}, {2}}.

From the ontological point of view it is important to ask what kind of objects sets are and whether they can be accepted by a nominalist. Mereological sets don’t create much of a problem, since they are just spatiotemporal objects, provided that their elements are spatiotemporal. The only contentious issue is whether we should admit the existence of arbitrary collections of objects. Is there an individual object that consists of my left pinkie, the planet Venus, and the left hind leg of some particular dinosaur? But the status of distributive sets is more controversial. Arguably, distributive sets cannot be identified with spatiotemporal objects. The singleton consisting of one physical object x is not identical with x, and because it cannot be identical with any other physical object y (since its existence would be contingent upon the existence of y, and the only acceptable ontological dependence of {x} is on x), hence {x} cannot be a physical object. Thus the most common interpretation of distributive sets is that they are abstract objects, and as such are not acceptable to the nominalist. However, nominalists can make use of the notion of distributive set in certain contexts, for which it is possible to give a nominalistic paraphrase. For instance, the statement “Socrates belongs to the set of all philosophers” can be interpreted nominalistically as “Socrates is a philosopher”. The sentence “The set of all philosophers is a subset of the set of all people” is interpreted as “All philosophers are humans”, and an analogous interpretation of the sentence “The set of all people is disjoint from the set of all elephants” can be given as “No humans are elephants”. Thus it can be claimed that the nominalist can accept first-order sets of physical objects (so-called classes). But higher-order sets, and especially those founded on the empty set, are not so easy to eliminate from the discourse.

At the beginning of the lecture we mentioned the fact that mathematical objects can be reduced to (distributive) sets. But, as Paul Benacerraf has famously noticed, such reductions are not unique. For instance, natural numbers can be interpreted as sets in at least two ways. One interpretation is given by following identifications: 0 = Ø, 1 = {Ø}, 2 = {Ø, {Ø}}, 3 = {Ø, {Ø}, {Ø, {Ø}}}, etc. But an alternative interpretation can look like this: 0 = Ø, 1 = {Ø}, 2 = {{Ø}}, 3 = {{{Ø}}}, etc. These two interpretations, taken literally, cannot be true, for this would imply mathematical falsehoods, such as {{Ø}} = {Ø, {Ø}} (you can prove that this identity is false, given the principle of extensionality and the assumption that {x} is different from x). But mathematical practice does not tell us which identification should be preferred. It looks like some questions regarding numerical identity between mathematical objects are fundamentally undecidable, which calls into question the ontological status of mathematical objects as independent entities. One solution to this problem is encompassed in the so-called structuralist interpretation of mathematics. According to this interpretation, the fundamental objects that mathematical theories speak about are whole structures, not individual objects. There is no number 1 as an entity that exists separately from the entire structure of natural numbers, hence it does not make sense to ask what this number is identical with. Mathematical objects are just positions in a given structure. There is more than one way to interpret one structure (e.g. the structure of natural numbers) within another structure (the structure of sets). But an interpretation is just a homomorphism, i.e. a mapping which preserves the structure. There is no identity involved. The essence of natural numbers is exhausted in the structure of a linear, discrete order.