Simplest Version of Godel Theorem

A while ago I read Smyllan's SELF version of Godel theorem which amounts to just a few lines.

The Liar paradox is universal in paradoxes. In fact there is only one Paradox and that is the Liar Paradox. It has been cleverly adapted to many, many situations but all are essentially the same. You wish to make a statement about the statement so that the statement can contradict itself. Since the contradiction is atomic you are stuck, there is no choice of statements to reject, you must either reject the whole statement making your system incomplete because by obeying the rule you make it True, or you just accept there are false statements making the system inconsistent. Godel's Theorem is so special because it formalises Paradox into symbolic logic providing a rigorous demonstration that is exists.

For SRH this is critical because it demonstrates the issues that lie at the basis of thinking in general. We often like to think that the universe and ourselves are fully rational and that everything is waiting to be uncovered. Or at least could be uncovered with enough searching. But if we ever tried to do this we would run into Paradox for the very reasons spelled out in the Liar Paradox and Godels Theorems.

Here's a simple version that captures Paradox perfectly. Credit to and quoted from Mark Dominus.

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.