What is Gödel's Incompleteness Theorem?

For a long time, mathematicians tried to prove that mathematics, as a logical system, is both consistent and complete. For mathematics to be consistent, it must not have any contradictions. For it to be complete, it must be shown that every true statement in mathematics can be proved as a theorem.

To accomplish this, mathematicians realized that mathematics must first be placed on a firm logical basis, and this was done by Alfred North Whitehead and Bertrand Russell when they published the three-volume book Principia Mathematica.

In 1931, Kurt Gödel surprised the whole world when he published a paper which became known as Gödel's Incompleteness Theorem. What Gödel essentially did in this paper was to use the symbolic logic constructed in Principia Mathematica to formulate a statement which, when translated to plain English, might read as follows:

Now, think about it. If this is a true statement, then can you prove it as a theorem? I guess not. How about if it's false? Well, then you shouldn't be able to prove it, because in order for mathematics to be consistent, you shouldn't be able to prove anything that's false. So, I guess it must be a true statement!

So there you have it! You have a true statement, but there is no way to prove it.

Gödel's Incompleteness Theorem is considered by many as the greatest accomplishment in mathematics in the 20th century. (In my opinion, the Turing Machine is only the second greatest accomplishment in mathematics in the 20th century, but it's there right behind Gödel's Incompleteness Theorem.)