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

This suggestion seems rather innocuous and perhaps even desirable, since,

after all, G asserts something true about the natural number system. But

what about the nonstandard type of extension? If it is at all parallel to the

case of the parallel postulate, it must involve

adding the negation of G as a new axiom.

But how can we even contemplate doing such a repugnant, hideous thing?

After all, to paraphrase the memorable words of Girolamo Saccheri, isn't

what -G says "repugnant to the nature of the natural numbers"?

Supernatural Numbers

I hope the irony of this quotation strikes you. The exact problem with

Saccheri's approach to geometry was that he began with a fixed notion of

what was true and what was not true, and he set out only to prove what he'd

assessed as true to start with. Despite t.he cleverness of his approach-which

involved denying the fifth postulate, and then proving many "repugnant"

propositions of the ensuing geometry-Saccheri never entertained the

possibility of other ways of thinking about points and lines. Now we should

be wary of repeating this famous mist.ake. We must consider impartially, to

the extent that we can, what it would mean to add -G as an axiom to TNT.

Just think what mathematics would be like today if people had never

considered adding new axioms of the following sorts:

3a:(a+a)=50

3a:5a=0

3a:(a· a) =550

3a:5(a·a) =0

While each of them is "repugnant to the nature of previously known
number systems", each of them also provides a deep and wonderful exten-
sion of the notion of whole numbers: rational numbers, negative numbers,
irrational numbers, imaginary numbers. Such a possibility is what -G is
trying to get us to open our eyes to. Now in the past, each new extension of
the notion of number was greeted with hoots and catcalls. You can hear this
particularly loudly in the names attached to the unwelcome arrivals, such as
"irrational numbers", "imaginary numbers". True to this tradition, we shall
name the numbers which -G is announcing to us the supernatural numbers,
.showing how we feel they violate all reasonable and commonsensical no-
tions.
If we are going to throw -G in as the sixth axiom of TNT, we had
better understand how in the world it. could coexist, in one system, with the
infinite pyramidal family we just finished discussing. To put it bluntly, -G
says:

452

"There exists some number which forms a TNT-proof-pair

with the arithmoquinification of u"

On Formally Undecidable Propositions