Now both of these contrasting types of sel f-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 G6del type, which assert their own unproducibility within specific formal sstems. 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 Godel sentences, the only difference being the omission of a negation.
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mentioned that a Henkin sentence tells nothing a bout 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 describe 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 0=0-a false statement, since 0=0 is not a derivation of anything. However, 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 derivation.
The essence of the distinction, then, between self-assembling units and non-self-assembling units is that the former get away with self-reproduction without telling the cell anything about their construction, while the latter need to give instructions as to how to assemble themselves. Now the parallel to Henkin sentences, implicit and explicit, ought to be quite clear. Implicit Henkin sentences are self-proving but do not tell anything at all about their proofs- they are analogous to self-assembling viruses; explicit Henkin sentences direct the construction of their own proofs-they are analogous to more complex viruses which direct their host cells in putting copies of themselves together.