Godel's proof, compressed

I found this here, but it is originally due to Raymond Smullyan. I first learned about Godel's theorem when I read Douglas Hofstader's book Godel, Escher, Bach in high school. I still remember lugging the thick volume around in my backpack, and working through the early chapters on predicate logic. This treatment is much more compact! :^)

We have some sort of machine that prints out statements in some sort of language. It needn't be a statement-printing machine exactly; it could be some sort of technique for taking statements and deciding if they are true. But let's think of it as a machine that prints out statements.

In particular, some of the statements that the machine might (or might not) print look like these:

P*x (which means that the machine will print x)
NP*x (which means that the machine will never print x)
PR*x (which means that the machine will print xx)
NPR*x (which means that the machine will never print xx)

For example, NPR*FOO means that the machine will never print FOOFOO. NP*FOOFOO means the same thing. So far, so good.

Now, let's consider the statement NPR*NPR*. This statement asserts that the machine will never print NPR*NPR*.

Either the machine prints NPR*NPR*, or it never prints NPR*NPR*.

If the machine prints NPR*NPR*, it has printed a false statement. But if the machine never prints NPR*NPR*, then NPR*NPR* is a true statement that the machine never prints.

So either the machine sometimes prints false statements, or there are true statements that it never prints.

So any machine that prints only true statements must fail to print some true statements.

Or conversely, any machine that prints every possible true statement must print some false statements too.