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

mechanisms for examining whether DNA is labeled or not, then the label

may make all the difference in the world. In particular, the host cell may

have an enzyme system· which looks for unlabeled DNA, and destroys any

that it finds by unmercifully chopping it to pieces. In that case, woe to all

unlabeled invaders!

The methyl labels on the nucleotides have been compared to serifs on

letters. Thus, using this metaphor, we could say that the E. coli cell is

looking for DNA written in its "home script", with its own particular

typeface-and will chop up any strand of DNA written in an "alien"

typeface. One counterstrategy, of course, is for phages to learn to label

themselves, and thereby become able to fool the cells which they are

invading into reproducing them.

This TC-battle can continue to arbitrary levels of complexity, but we

shall not pursue it further. The essential fact is that it is a battle between a

host which is trying to reject all invading DNA, and a phage which is trying

to infiltrate its DNA into some host which will transcribe it into mRNA

(after which its reproduction is guaranteed). Any phage DNA which suc-

ceeds in getting itself reproduced this way can be thought of as having this

high-level interpretation: "I Can Be Reproduced in'Cells of Type X". This

is to be distinguished from the evolutionarily pointless kind of phage

mentioned earlier, which codes for proteins that destroy it, and whose

high-level interpretation is the self-defeating sentence: "I Cannot Be Re-

produced in Cells of Type X".

Henkin Sentences and Viruses

Now both of these contrasting types of self-reference in molecular biology

have their counterparts in mathematical logic. We have already discussed

the analogue of the self-defeating phages-namely, strings of the Codel

type, which assert their own unproducibility within specific formal systems.

But one can also make a counterpart sentence to a real phage: the phage

asserts its own producibility in a specific cell, and the sentence asserts its

own producibility in a specific formal system. Sentences of this type are

called Henkin sentences, after the mathematical logician Leon Henkin. They

can be constructed exactly along the lines of Codel sentences, the only

difference being the omission of a negation. One begins with an "uncle", of

course:

3a:3a':<TNT-PROOF-PAIR{a,a'}!\ARITHMOQUINE{a",a'}>

and then proceeds by the standard trick. Say the Code! number of the

above "uncle" is h. Now by arithmoquining this very uncle, you get a

Henkin sentence:

3a:3a':<TNT-PROOF-PAIR{a,a'}!\ARITHMOQUINE{SSS ... SSSO/a",a'}>

----------h S's

Self-Ref and Self-Rep 541