Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter

We can dignify this observation by calling it the Central Dogma of

Mathematical Logic, and depicting it in a two-step diagram:

TNT ~ N ~ meta-TNT

In words: a string of TNT has an interpretation in N; and a statement of N

may have a second meaning as a statement about TNT.

G: A String Which Talks about Itself in Code

This much is intriguing yet it is only half the story. The rest of the story
involves an intensification of the self-reference. We are now at the stage
where the Tortoise was when he realized that a record could be made
which would make the phonograph playing it break-but now the question
is: "Given a record player, how do you actually figure out what to put on
the record?" That is a tricky matter.
We want to find a string of TNT-which we'll call 'G'-which is about
itself, in the sense that one of its passive meanings is a sentence about G. In
particular the passive meaning will turn out to be

"G is not a theorem of TNT."

I should quickly add that G also has a passive meaning which is a statement of
number theory; just like MUMON it is susceptible to being construed in (at
least) two different ways. The important thing is that each passive meaning
is valid and useful and doesn't cast doubt on the other passive meaning in
any way. (The fact that a phonograph playing a record can induce vibra-
tions in itself and in the record does not diminish in any way the fact that
those vibrations are musical sounds!)

G's Existence Is What Causes TNT's Incompleteness

The ingenious method of creating G, and some important concepts relat-

ing to TNT, will be developed in Chapters XIII and XIV; for now it isjust

interesting to glance ahead, a bit superficially, at the consequences of

finding a self-referential piece of TNT. Who knows? It might blow up! In a

sense it does. We focus down on the obvious question:

Is G a theorem of TNT, or not?

Let us be sure to form our own opinion on this matter, rather than rely on

G's opinion about itself. After all, G may not understand itself any better

than a Zen master understands himself. Like MUMON, G may express a

falsity. Like MU, G may be a nontheorem. We don't need to believe every

possible string of TNT-only its theorems. Now let us use our power of

reasoning to clarify the issue as best we can at this point.

We will make our usual assumption: that TNT incorporates valid

Murnon and Godel^271