Godel's Theorem in One Sentence

Godel's Theorem shows that there exists a very long logical string that is true but which cannot possibly be a theorem because if it were a theorem then its Godel number, say z, would stand in a certain relation to the Godel number of its proof, contrary to the alleged theorem itself which can only be interpreted to mean that no number stands in this relation to z.


To visualize this, imagine a huge string of logic (that I'll call the "second string") that begins "For all k, for all m, there exists y such that for all j and n...." and goes on and on and on---a sentence which has a fantastically large Godel number by the way---and then also imagine an even bigger string (that I'll call the "first string") that is the exact proof of the second string, but which has an even more fantasticulistically large Godel number. Also understand that there is an exact arithmetic relation between these two numbers (it so happens, because one is the derivation of the other) and that the second string, the theorem, means precisely that for no number can there be this exact such arithmetic relation between it and the first string, the derivation.

So what's true about numbers isn't identical to what can be formally proved about them within a fixed formal system of axioms.

If you've no books on Godel's Theorem, you must start, for about ten dollars, with Nagel and Newman's small paperback entitled "Godel's Proof", and make sure that you memorize the phrase which they use over and over again that says what sub(y, 13, y) means.