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

-but the various members of the pyramidal family successively assert:

"0 is not that number"

"1 is not that number"

"2 is not that number"

This is rather confusing, because it seems to be a complete contradiction

(which is why it is called "w-inconsistency"). At the root of our

confusion-much as in the case of the splitting of geometry-is our stub-

born resistance to adopt a modified interpretation for the symbols, despite

the fact that we are quite aware that the system is a modified system. We

want to get away without reinterpreting any symbols-and of course that

will prove impossible.

The reconciliation comes when we reinterpret 3 as "There exists a

generalized natural number", rather than as "There exists a natural

number". As we do this, we shall also reinterpret V in the corresponding

way. This means that we are opening the door to some extra numbers

besides the natural numbers. These are the supernatural numbers. The

naturals and supernaturals together make up the totality of generalized

naturals.

The apparent contradiction vanishes into thin air, now, for the pyram-

idal family still says what it said before: "No natural number forms a

TNT -proof-pair with the arithmoquinification of u." The family doesn't

say anything about supernatural numbers, because there are no numerals

for them. But now, -G says, "There exists a generalized natural number

which forms a TNT-proof-pair with the arithmoquinification of u." It is

clear that taken together, the family and -G tell us something: that there is

a supernatural number which forms a TNT-proof-pair with the arithmo-

quinification of u. That is all-there is no contradiction any more.

TNT +-G is a consistent system, under an interpretation which includes

supernatural numbers.

Since we have now agreed to extend the interpretations of the two

quantifiers, this means that any theorem which involves either of them has

an extended meaning. For example, the commutativity theorem

Va:Va':(a+a')=(a' +a)

now tells us that addition is commutative for all generalized natural

numbers-in other words, not only for natural numbers, but also for

supernatural numbers. Likewise, the TNT-theorem which says "2 is not the

square of a natural number"-

-3a:(a ·a)=SSO

-now tells us that 2 is not the square of a supernatural number, either. In

fact, supernatural numbers share all the properties of natural numbers, as

On Formally Undecidable Propositions 453