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

which can be derived in the same way as was derived.
Before we give more rules, let us give the five axioms of TNT:

AXIOM 1: Va:-Sa=O

AXIOM 2: Va:(a+O)=a

AXIOM 3: Va:Vb:(a+Sb)=S(a+b)

AXIOM 4: Va:(a'O)=O

AXIOM 5: Va:\fb:(a'Sb)=((a'b)+a)

(In the austere versions, use a' instead of b.) All of them are very simple to
understand. Axiom 1 states a special fact about the number 0; Axioms 2
and 3 are concerned with the nature of addition; Axioms 4 and 5 are
concerned with the nature of multiplication, and in particular with its
relation to addition.

The Five Peano Postulates

By the way, the interpretation of Axiom 1-"Zero is not the successor of
any natural number"-is one of five famous properties of natural numbers
first explicitly recognized by the mathematician and logician Giuseppe
Peano, in 1889. In setting out his postulates, Peano was following the path
of Euclid in this way: he made no attempt to formalize the principles of
reasoning, but tried to give a small set of properties of natural numbers
from which everything else could be derived by reasoning. Pea no's attempt
might thus be considered "semiformal". Pea no's work had a significant
influence, and thus it would be good to show Peano's five postulates. Since
the notion of "natural number" is the one which Peano was attempting to
define, we will not use the familiar term "natural number", which is laden
with connotation. We will replace it with the undefined term djinn, a word
which comes fresh and free of connotations to our mind. Then Pea no's five
postulates place five restrictions on djinns. There are two other undefined
terms: Genie, and meta. I will let you figure out for yourself what usual
concept each of them is supposed to represent. The five Peano postulates:

( 1) Genie is a d jinn.

(2) Every djinn has a meta (which is also a djinn).

(3) Genie is not the meta of any djinn.

(4) Different djinns have different metas.

(5) If Genie has X, and each djinn relays X to its meta, then all

djinns get X.

In light of the lamps of the Little Harmonic Labyrinth, we should name the set

of all djinns "GOD". This harks back to a celebrated statement by the

German mathematician and logician Leopold Kronecker, archenemy of

Georg Cantor: "God made the natural numbers; all the rest is the work of

man."

(^216) Typographical Number Theory