long as those properties are given to us in theorems of TNT. In other
words, everything that can be formally proven about natural numbers is
thereby established also for supernatural numbers. This means, in particu-
lar, that supernatural numbers are not anything already familiar to you,
such as fractions, or negative numbers, or complex numbers, or whatever.
The supernatural numbers are, instead, best visualized as integers which
are greater than all natural numbers--as infinitely large integers. Here is the
point: although theorems of TNT can rule out negative numbers, frac-
tions, irrational numbers, and complex numbers, still there is no way to
rule out infinitely large integers. The problem is, there is no way even to
express the statement "There are no infinite quantities".
This sounds quite strange, at first. Just exactly how big is the number
which makes a TNT-proof-pair with G's Godel number? (Let's call it 'I',
for no particular reason.) Unfortunately, we have not got any good vocabu-
lary for describing the sizes of infinitely large integers, so I am afraid I
cannot convey a sense of I's magnitude. But then just how big is i (the
square root of -I)? Its size cannot be imagined in terms of the sizes of
familiar natural numbers. You can't say, "Well, i is about half as big as 14,
and 9/10 as big as 24." You have to say, "i squared is -I", and more or less
leave it at that. A quote from Abraham Lincoln seems a propos here.
When he was asked, "How long should a man's legs be?" he drawled, "Long
enough to reach the ground." That is more or less how to answer the
question about the size of I-it should be just the size of a number which
specifies the structure of a proof of G-no bigger, no smaller.
Of course, any theorem of TNT has many different derivations, so you
might complain that my characterization of I is nonunique. That is so. But
the parallel with i-the square root of -I-still holds. Namely, recall that
there is another number whose square is also minus one: -i. Now i and -i
are not the same number. They just have a property in common. The only
trouble is that it is the property which defines them! We have to choose one
of them-it doesn't matter which one·-and call it "i". In fact there's no way
of telling them apart. So for all we know we could have been calling the
wrong one "i" for all these centuries and it would have made no difference.
Now, like i, I is also nonuniquely defined. So you just have to think of I as
being some specific one of the many possible supernatural numbers which
form TNT -proof-pairs with the arithmoquinification of u.
Supernatural Theorems Have Infinitely Long DerivationsWe haven't yet faced head on what it means to throw -G in as an axiom.
We have said it but not stressed it. The point is that -G asserts that G has a
proof How can a system survive, when one of its axioms asserts that its own
negation has a proof? We must be in hot water now! Well, it is not so bad as
you might think. As long as we only construct finite proofs, we will never
prove G. Therefore, no calamitous collision between G and its negation -G
will ever take place. The supernatural number I won't cause any disaster.(^454) On Formally Undecidable Propositions