I Am a Strange Loop

While you are pondering this, I will jump back to the source of it all, which was Gödel’s PM formula that talked about itself. The point is that Gödel numbers, since they can be used as names for formulas and can be inserted into formulas, are precisely analogous to quoted phrases. Now we have just seen that there is a way to use quotation marks and sentence fragments to make a full sentence that talks about itself (or if you prefer, a sentence that talks about another sentence, but one that is a clone to it, so that whatever is true of the one is true of the other).

Gödel, analogously, created a “subjectless formula fragment” (by which I mean a PM formula that is not about any specific integer, but just about some unspecified variable number x). And then, making a move analogous to that of feeding Quine’s Quasi-Quip into itself (but in quotes), he took that formula fragment’s Gödel number k (which is a specific number, not a variable) and replaced the variable x by it, thus producing a formula (not just a fragment) that made a claim about a much larger integer, g. And g is the Gödel number of that very claim. And last but not least, the claim was not about whether the entity in question was a full sentence or not, but about whether the entity in question was a provable formula or not.

An Elephant in a Matchbox is Neither Fish Nor Fowl

I know this is a lot to swallow in one gulp, and so if it takes you several gulps (careful rereadings), please don’t feel discouraged. I’ve met quite a few sophisticated mathematicians who admit that they never understood this argument totally!

I think it would be helpful at this juncture to exhibit a kind of hybrid sentence that gets across the essential flavor of Gödel’s self-referential construction but that does so in Quinean terms — that is, using the ideas we’ve just been discussing. The hybrid sentence looks like this:

“when fed its own Gödel number yields a non-prim number”

when fed its own Gödel number yields a non-prim number.

The above sentence is neither fish nor fowl, for it is not a formula of Principia Mathematica but an English sentence, so of course it doesn’t have a Gödel number and it couldn’t possibly be a theorem (or a nontheorem) of PM. What a mixed metaphor!

And yet, mixed metaphor though it is, it still does a pretty decent job of getting across the flavor of the PM formula that Gödel actually concocted. You just have to keep in mind that using quote marks is a metaphor for taking Gödel numbers, so the upper line should be thought of as being a Gödel number (k) rather than as being a sentence fragment in quote marks. This means that metaphorically, the lower line (an English sentence fragment) has been fed its own Gödel number as its subject. Very cute!

I know that this is very tricky, so let me state it once again, slightly differently. Gödel asks you to imagine the formula that k stands for (that formula happens to contain the variable x), and then to feed k into it (this means to replace the single letter x by the extremely long numeral k, thus giving you a much bigger formula than you started with), and to take the Gödel number of the result. That will be the number g, huger far than k — and lastly, Gödel asserts that this walloping number is not a prim number. If you’ve followed my hand-waving argument, you will agree that the full formula’s Gödel number (g) is not found explicitly inside the formula, but instead is very subtly described by the formula. The elephant’s DNA has been used to get a description of the entire elephant into the matchbox.

Sluggo and the Morton Salt Girl

Well, I don’t want to stress the technical points here. The main thing to remember is that Gödel devised a very clever number-description trick — a recipe for making a very huge number g out of a less huge number k — in order to get a formula of PM to make a claim about its own Gödel number’s non-primness (which means that the formula is actually making a claim of its own nontheoremhood). And you might also try to remember that the “little” number k is the Gödel number of a “formula fragment” containing a variable x, analogous to a subjectless sentence fragment in quote marks, while the larger number g is the Gödel number of a complete sentence in PM notation, analogous to a complete sentence in English.

Popular culture is by no means immune to the delights of self-reference, and it happens that the two ideas we have been contrasting here — having a formula contain its own Gödel number directly (which would necessitate an infinite regress) and having a formula contain a description of its Gödel number (which beautifully bypasses the infinite regress) — are charmingly illustrated by two images with which readers may be familiar.