However, we will have to get used to the idea that -G is now the one which
asserts a truth ("G has a proof"), while G asserts a falsity ("G has no
proof"). In standard number theory it is the other way around-but then,
in standard number theory there aren't any supernatural numbers. Notice
that a supernatural theorem of TNT-namely G-may assert a falsity, but
all natural theorems still assert truths.Supernatural Addition and MultiplicationThere is one extremely curious and unexpected fact about supernaturals
which I would like to tell you, without proof. (I don't know the proof
either.) This fact is reminiscent of the Heisenberg uncertainty principle in
quantum mechanics. It turns out that you can "index" the supernaturals in
a simple and natural way by associating with each supernatural number a
trio of ordinary integers (including negative ones). Thus, our original
supernatural number, I, might have the index set (9,-8,3), and its succes-
sor, 1+ 1, might have the index set (9,-8,4). Now there is no unique way to
index the supernaturals; different methods offer different advantages and
disadvantages. Under some indexing schemes, it is very easy to calculate
the index triplet for the sum of two supernaturals, given the indices of the
two numbers to be added. Under other indexing schemes, it is very easy to
calculate the index triplet for the product of two supernaturals, given the
indices of the two numbers to be multiplied. But under no indexing scheme
is it possible to calculate both. More precisely, if the sum's index can be
calculated by a recursive function, then the product's index will not be a
recursive function; and conversely, if the product's index is a recursive
function, then the sum's index will not be. Therefore, supernatural school-
children who learn their supernatural plus-tables will have to be excused if
they do not know their supernatural times-tables-and vice versa! You
cannot know both at the same time.Supernaturals Are Useful ...
One can go beyond the number theory of supernaturals, and consider
supernatural fractions (ratios of two supernaturals), supernatural real
numbers, and so on. In fact, the calculus can be put on a new footing, using
the notion of supernatural real numbers. Infinitesimals such as dx and dy,
those old bugaboos of mathematicians, can be completely justified, by
considering them to be reciprocals of infinitely large real numbers! Some
theorems in advanced analysis can be proven more intuitively with the aid
of "nonstandard analysis".But Are They Real?Nonstandard number theory is a disorienting thing when you first meet up
with it. But then, non-Euclidean geometry is also a disorienting subject. In
On Formally Undecidable Propositions^455