(By the way, can you spot how this sentence differs from -G?) The reason I
show it explicitly is to point out that a Henkin sentence does not give a full
recipe for its own derivation; it just asserts that there exists one. You might
well wonder whether its claim is justified. Do Henkin sentences indeed
possess derivations? Are they, as thev claim, theorems? It is useful to recall
that one need not believe a politician who says, "I am honest"-he may be
honest, and yet he may not be. Are Henkin sentences any more trustworthy
than politicians? Or do Henkin sentences, like politicians, lie in cast-iron
sinks?
It turns out that these Henkin sentences are invariably truth tellers.
Why this is so is not obvious; but we will accept this curious fact without
proof.
Implicit vs. Explicit Henkin Sentences
I mentioned that a Henkin sentence tells nothing about its own derivation;
it just asserts that one exists. Now it is possible to invent a variation on the
theme of Henkin sentences-namely sentences which explicitly describe their
own derivations. Such a sentence's high-level interpretation would not be
"Some Sequence of Strings Exists Which is a Derivation of Me", but rather,
"The Herein-described Sequence of Strings ..... Is a Derivation of Me".
Let us call the first type of sentence an implicit Henkin sentence. The new
sentences will be called explicit Henkin sentences, since they explicitly de-
scribe their own derivations. Note that, unlike their implicit brethren,
explicit Henkin sentences need not be theorems. In fact, it is quite easy to write
a string which asserts that its own derivation consists of the single string
O=O-a false statement, since 0=0 is not a derivation of anything. How-
ever, it is also possible to write an explicit Henkin sentence which is a
theorem-that is, a sentence which in fact gives a recipe for its own deriva-
tion.Henkin Sentences and Self-AssemblyThe reason I bring up this distinction between explicit and implicit Henkin
sentences is that it corresponds very nicely to a significant distinction
between types of virus. There are certain viruses, such as the so-called
"tobacco mosaic virus", which are called self-assembling viruses; and then
there are others, such as our favorite T-evens, which are non-self-assembling.
Now what is this distinction? It is a direct analogue to the distinction
between implicit and explicit Henkin sentences.
The DNA of a self-assembling \irus codes only for the parts of a new
virus, but not for any enzymes. Once the parts are produced, the sneaky
virus relies upon them to link up to each other without help from any
enzymes. Such a process depends on chemical affinities which the parts
have for each other, when swimming in the rich chemical brew of a cell.
Not only viruses, but also some organelles-such as ribosomes-assemble(^542) Self-Ref and Self-Rep