7What sets are there?

You might wonder which sets, exactly, exist. The answer to this question is somewhere between “complicated” and “unknown”. But for the kinds of sets we will be concerned with, it will be fine for us to tacitly apply what is sometimes called the naive comprehension principle:

You might have heard that Bertrand Russell showed that this principle (or rather, a closely related principle due to Gottlob Frege, known as Basic Law V) is inconsistent. That is true, the axiom of naive comprehension is inconsistent, and this is exactly why the question of which sets exist gets complicated.

Mathematicians have come up with sets of axioms which give us rules for when sets exist, the most famous of which are called the Zermelo–Fraenkel axioms. Unlike the naive comprehension principle, these axioms are probably consistent, but they are complicated, and they need not bother us. For our purposes, it will be fine to act as though the naive comprehension principle is true, and that any sets we can think of really do exist.