For a long time, the theoretical basis for mathematics worked with Cantor's theory of sets. One of his axioms stated that any logical condition featuring an element already determines a set, namely the set of all elements that make this condition true. What can go wrong with such an idea? Consider the following set. We can form it in just any universe, for instance in the universe of real numbers.

A = {a a real number; a∉A}.

What are elements of this sets? We take any real number a and try to determine whether it belongs to A or not. If it does belong to A, then the condition in the description of this set si false, which means that it actually should not be in A. But if it is not in this set, then the condition is true and it should be in A.

This logical vicious circle shows that we are unable to actually decide what elements are in this set. This is the famous Russell paradox (from 1901) that meant the end of intuitive (or naive) set theory. Obviously, if we want to avoid this kind of trouble, we have to be more careful in defining sets. A rigorous set theory was developed, it places reasonable restrictions on what kinds of definition one can use (in fact there are more theories of sets, each trying to eliminate possibility of paradoxes in its own way). Of course, this is way beyond the scope of Math Tutor.