The set theory taught to us in the secondary school levels has a big trouble — It cannot be called a theory in the first place. For it suffers a contradiction, viz.,
Let us call a set “abnormal” if it is a member of itself, and “normal” otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is “normal”. On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is “abnormal”.
Now we consider the set of all normal sets – let us give it a name: R. If R were abnormal, that is, if R were a member of itself, then since R only contains normal sets, R must be normal, which is contradictory to our original hypothesis: R is abnormal. So, R cannot be abnormal, which means R is normal. Further, since every normal set is a member of R, R itself must be a member of R, making R abnormal. Paradoxically, we are led to the contradiction that R is both normal and abnormal.
This paradox is, often, referred to as Russell’s Paradox. Fortunately, consistent set theories exist — One of them is Zermelo–Fraenkel set theory.
Coming back to the informal set theories taught to us — these are attributed to the legendary Georg Cantor. Besides set theory, he gave us the wonderful notion of countable sets, uncountable sets and infinities. In a revolutionary result, he claims the following–
Not all infinities are of the same size! He, also, claims - Give me an infinity and I shall construct an infinity bigger than yours!
Informally, he claims In particular, the power set of a countably infinite set is uncountably infinite.
There are few more interesting things that come out of Cantor’s Results –
- The ratio number of problems that can be solved on a computer to those that cannot be solved is exactly equal to the ratio of number of Natural Numbers to Real Numbers.
- He, also, goes on to prove that set of real numbers is the smallest infinity that one can find.
- The number of Real numbers between 0 and 1 is exactly equal to the total number of Real numbers!
