1. Let us consider an object, where by object is meant that which is distinct from everything else, that which has an identity, that which is nothing but itself. And furthermore, for the moment, let us suppose that all there is are objects.

2. This implies that there is at least one other object, since otherwise the above object is not distinct, since there is nothing to distinguish it from.

3. Thus, all objects are distinct and distinguishable. Furthermore, since they are all objects, then they are identical as objects. Thus, all objects are both distinct and identical. Thus, each of these objects may be written as S1, S2, ... and all of these objects may be written as S. S is called a set, although we will mean by set only that which is defined in these notes.

4. Now, does S have an identity; is it what it is and nothing else? If all there are are objects then S has no identity since it is not distinct from anything else. We can't even say what it is, that is, only objects, even that it is one; we can't even name it. The very naming of S distinguishes something from that which it is not. That is, the naming or claiming something as S is equivalently a statement that it is not (not S).

5. Furthermore, even if S did have an identity, a meaning, S is not an object, but a number of, or perhaps a collection of objects. This is in contradiction of our assumption. If there are entities other than objects, such as S, then there can be no "ultimate" kind of entity. Thus, if there is any kind of entity, there is no ultimate kind of entity.

6. All objects therefore, are both distinct from each other and identical to each other; we call all of them a set. The set itself, in having an identity, must be distinct from its negations, its sets of non-objects. At the same time, it must be identical to the sets of non-objects, etc.

7. It is possible to have things with no identity, but all identical. The collection of these things is called a sort. If there is more than one kind of sort, each has an identity and they are then objects.