Godelís incompleteness theorem
If you believed everything you read about Godelís incompleteness theorem, you could be forgiven for thinking it holds the key to life, the universe and everything. By some accounts, this equation explains the human mind, it determines the nature of free will and it limits what we can ultimately know about the universe.
But there is really no need for hype. Mathematicians generally agree that Godelís incompleteness theorem is one of the most important achievements in mathematics. It is also one of the most mysterious and disturbing.
At the end of the 19th century mathematicians had huge optimism that a new kind mathematics was emerging that would eventually explain everything the Universe could throw at it. In 1920, the German mathematician David Hilbert proposed a research program that would aim to build all of mathematics on solid and complete mathematical foundations. He believed this process would show that mathematics contained no contradictions, that in mathematical language, it was consistent.
Hilbert was wrong. And the man who proved it was Kurt Godel.
Godel discovered a property of any logical system that truly astounded mathematicians. He began by thinking about the way rules can be used to make statements. What Godel found was that if these rules contain no contradictions then there is something strange about the statements that can be made with them: certain statements cannot be proved true using the available rules, even though they are true. Instead extra rules are needed to prove the point. So the original set of rules must be incomplete.
Godelís theorem is that if a set of rules are consistent, they are incomplete. For example, arithmetic is a set of rules for making statements about numbers. These rules contain no contradictions and so by Godel's theorem must be incomplete.
At a stroke, the infallibility of mathematics was shattered. The theorem means that it is not possible to construct solid and complete foundations for mathematics in a way that allows all mathematical truths to be proved.
The philosophical implications of this are hotly debated. Many people have wondered whether the laws of physics are a consistent set of rules. If so, Godelís theorem would apply meaning they must be incomplete. But little headway has been made in the process of formulating the laws of physics in a consistent way and there is little agreement on whether it is even possible. Could there exist laws of physics that are true and yet unprovable? Maybe.
Others have wondered what Godelís theorem means for our understanding of the human mind. If our brains are machines that work in a consistent way, then Godelís theorem applies. Does that mean that it is possible to think of ideas that are true but be unable to prove them? Nobody knows.
A discussion of Godel's incompleteness theorems is at
See also Godel's Proof, Ernest Nagel and James Newman, Routledge