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

will have to Godel-number TNT itself, just as we did the MIU-system, and

then "arithmetize" its rules of inference. The Godel-numbering is easy to

do. For instance, we could make the following correspondence:

Symbol

o s + < > [ ] a

f\

V

::l

3

'V

pune.

Codon

666

123

111

112

236

362

323

212

213

312

313

262

163

161

616

633

223

333

626

636

611

Mnemonic Justification

Number of the Beast for the Mysterious Zero

successorship: 1, 2, 3, ...

visual resemblance, turned sideways

1 + 1 = 2

2 x 3 = 6

ends in 2

ends in 3

ends in 2

ends in 3

ends in 2

ends in 3

these three pairs

form a pattern

opposite to 'V (626)

163 is prime

'1\' is a "graph" of the sequence 1-6-1

V is a "graph" of the sequence 6-1-6

6 "implies" 3 and 3, in some sense ...

2 + 2 is not 3

'3' looks like '3'

opposite to a; also a "graph" of 6-2-6

two dots, two sixes

special number, as on Bell system (411, 911)

Each symbol of TNT is matched up with a triplet composed of the
digits 1,2,3, and 6, in a manner chosen for mnemonic value. I shall call
each such triplet of digits a Gadel codon, or codon for short. Notice that I
have given no codon for b, C, d, or e; we are using austere TNT. There is a
hidden motivation for this, which you will find out about in Chapter XVI.
I will explain the bottom entry, "punctuation", in Chapter XIV.
Now we can rewrite any string or rule of TNT in the new garb. Here,
for instance, is Axiom 1 in the two notations, the old below the new:
626,262,636,223,123,262, 111,666
'VaS a 0

Conveniently, the standard convention of putting in a comma every third
digit happens to coincide with our codons, setting them off for "easy"
legibility.
Here is the Rule of Detachment, in the new notation:

RULE: If x and 212x633y213 are both theorems, then y is a theorem.

Finally, here is an entire derivation taken from last Chapter, given in
austere TNT and also transcribed into the new notation:

(^268) Murnon and Godel