Set

If we start with nothing at all, we should be able to agree that there exists a set that contains nothing at all, aka the empty set. The empty set has the definite article because it is unique - there's no difference between sets that are empty, so there is just one such set, the empty set. While, it may seem easy enough to define the empty set, it is still just a single instance of a set, so we'd have to unite all other types (and they actually contain something) in the final set definition. However, it turns out that defining a set cannot be done consistently, in which case we introduce the concept of a set as a mathematical primitive.

Mathematical primitives have the no-questions-ask priviledge and they're introduces by appealing to readers' intuition. In other words, primitives are just given by a text author to be taken for granted by readers. A primitive is created by the mathematical magic wand proper that conjures things up out of nothing, just by appealing to one's inuition. The authors should try their best to describe what a set is (or some other primitive) before playing the primitive card, for it's actually them authors that create the idea of a set by means of painting a particular picture in their readers' minds.