How Do Sets Have Members?

Sets are collections of things. The things of a set may be of all varieties–from numbers, to letters, to objects, to sets themselves.

How do sets get the members they do have, while keeping out those things that aren’t their members?

The answer to this question lies with the things themselves–any combination of things at all is a set, and any difference in combination is a difference in set. An additional answer concerns how we determine combinations of things by way of sets. After all, if we are to use sets, we need to know about the sort of things we are (or are presumably) dealing with. The answer, I will explain, is that sets specify their members, while the things not specified are not their members. So how do sets specify their members?

I’ve seen this done in one of two ways: either by what I’ll call naming, or by giving what I’ll term a property that the thing must have.

For naming, the set simply provides the name, or lists the names so that the things referred to by the names are to be included as members of the set. For example,

A = { 1, 2, 3 }

Here, set A has as members the numbers 1, 2, and 3, by naming them. It could have included these same numbers via a different name, perhaps by the Roman Numerals I, II, and III, as opposed to their Arabic counterparts.

For the other way of having members, the set gives a property so that the things that have the property are to be included as members of the set. For example,

B = { xx = x }

This says that B is the set of anything in place of x such that x is identical with x. And so, set B has as a member anything at all that satisfies the property of being identical with itself. Since this property is satisfied by each and every thing, everything is a member of B.

A property that nothing has, or even a property that constitutes a contradiction, may be given by a set. For example,

C = { xx ≠ x }

D = { x | (x = 1 & x ≠ 1) }

Set C gives a property that nothing satisfies, since nothing is not identical with itself. Set D also gives a property that nothing satisfies, because nothing has a property that leads to a contradiction. Since C and D give properties that nothing satisfies, C and D have no members at all. They are what are called null sets. If sets are to be identified by their members (this is called the Axiom of Extension), then C and D are one and the same: the null set, typically denoted with ∅.

I put this later method of a set’s having members in term of properties. However, the same point may be made in terms of a test that the thing passes, or a condition that the thing meets. It should make no difference at all whether we conceive of things as being members of a set by having the property the set gives, or by passing the test the set gives, or by meeting the condition the thing gives. It’s really all the same.

Perhaps these two ways I’ve presented need not exhaust all the ways that things can be members of sets. However, these are the only ways I’ve found in the literature on set theory, and the only ways I know about.

One thing I have not yet covered in this post is the possibility of putting restrictions on which sort of properties can be specified by sets. In particular, Russell’s Paradox is considered to require a restriction on the kinds of properties that sets may specify. That is, while sets still do specify properties in order to have the members they have, perhaps not every property is fair game. I will leave this subject for a later post.