methods of reasoning, and therefore that TNT never has falsities for
theorems. In other words, anything which is a theorem of TNT expresses a
truth. So if G were a theorem, it would express a truth, namely: "G is not a
theorem". The full force of its self-reference hits us. By being a theorem, G
would have to be a falsity. Relying on our assumption that TNT never has
falsities for theorems, we'd be forced to conclude that G is not a theorem.
This is all right; it leaves us, however, with a lesser problem. Knowing that
G is not a theorem, we'd have to concede that G expresses a truth. Here is a
situation in which TNT doesn't live up to our expectations-we have found
a string which expresses a true statement yet the string is not a theorem.
And in our amazement, we shouldn't lose track of the fact that G has an
arithmetical interpretation, too-which allows us to summarize our
findings this way:
A string of TNT has been found; it expresses, unambiguously, a
statement about certain arithmetical properties of natural num-
bers; moreover, by reasoning outside the system we can determine
not only that the statement is a true one, but also that the string
fails to be a theorem of TNT. And thus, if we ask TNT whether
the statement is true, TNT says neither yes nor no.Is the Tortoise's string in the Mu Offering the analogue ofG? Not quite.
The analogue of the Tortoise's string is -G. Why is this so? Well, let us
think a moment about what -G says. It must say the opposite of what G
says. G says, "G is not a theorem of TNT", so -G must say "G is a theorem".
We could rephrase both G and -G this way:
G: "I am not a theorem (of TNT)."
-G: "My negation is a theorem (of TNT)."It is -G which is parallel to the Tortoise's string, for that string spoke not
about itself, but about the string which the Tortoise first proffered to
Achilles-which had an extra knot on it (or one too few, however you want
to look at it).Murnon Has the Last WordMumon penetrated into the Mystery of the Undecidable as clearly as
anyone, in his concise poem on J6shli's MU:272
Has a dog Buddha-nature?
This is the most serious question of all.
If you say yes or no,
You lose your own Buddha-nature.Murnon and G6del