In the previous chapter I presented the core ideas of set theory. The exposition was not very formal, nevertheless, it was rigorous enough for our purposes. It is easy to see that set theory assumes that there is no vagueness. However, in the first chapter, I introduced vagueness and explained why vagueness is not just linguistic phenomenon. The real question is if we can define some sort of alternative mathematics where vagueness plays a central role. In what follows, I will explain what what is alternative mathematics and how we can introduce vague mathematics (and not mathematics of vagueness).