Sets are at the foundation of modern mathematics. However, many problems arise with the introduction of Axiomatic Set Theory. Not to put too fine a point on it, the following are examples.

Does the set of all sets contain itself as a member?

Consider the set S of all sets that aren't members of themselves. Is S a member of itself?

And to confuse membership, consider all the things in a given refrigerator. Postulate the absence of pineapples. In some sense, the absence of pineapples is a presence in the refrigerator. If you're hungry for pineapples, you notice the absence of pineapples in the refrigerator. "No pineapples" are a presence in the set of things in the refrigerator. This is not simply the empty set. It is a way of distinguishing between different qualities of absence.