I Am a Strange Loop

But suppose someone told you that they had built an analogous machine — I’ll dub it “Göru” — that would always correctly answer any question of the form “Is n a prim number?” Would this claim — strictly analogous to the previous one — strike you as equally ho-hum? If so, then I respectfully submit that you’ve got another think coming.

The reason is this. If you believed Göru to be reliable and you also believed in the Mathematician’s Credo (Principia Mathematica version), then you could conclude that your little Göru, working all by itself, could answer any number-theoretical question that you were interested in, just like a genie conjured from a magic lamp. How so? What makes Göru a magic genie?

Well, suppose you wanted to know if statement X is true or false (for instance, the famous claim “Every even number greater than 2 is the sum of two primes” — which, as I stated above, remains unsettled even today, after nearly three centuries of work). You would just write X down in the formal notation of PM, then convert that formula mechanically into its Gödel number x, and feed that number into Göru (thus asking if x is prim or not). Of course x will be a huge integer, so it would probably take Göru a good while to give you an answer, but (assuming that Göru is not a hoax) sooner or later it would spit out either a “yes” or a “no”. In case Göru said “yes”, you would know that x is a prim number, which tells you that the formula it encodes is a provable formula, which means that statement X is true. Conversely, were Göru to tell you “no”, then you would know that the statement X is not provable, and so, believing in the Mathematician’s Credo (Principia Mathematica version), you would conclude it is false.

In other words, if we only had a machine that could infallibly tell apart prim numbers and “saucy” (non-prim) numbers, and taking for granted that the Principia Mathematica version of the Mathematician’s Credo is valid, then we could infallibly tell true statements from false ones. In short, having a Göru would give us a royal key to all of mathematical knowledge.

The prim numbers alone would therefore seem to contain, in a cloaked fashion, all of mathematical knowledge wrapped up inside them! No other sequence of numbers ever dreamt up by anyone before Gödel had anything like this kind of magically oracular quality. These amazing numbers seem to be worth their weight in gold! But as I told you, the prim numbers are elusive, because small ones sometimes wind up being added to the club at very late stages, so it won’t be easy to tell prim numbers from saucy ones, nor to build a Göru. (This is meant as a premonition of things to come.)

Gödelian Strangeness

Finally, Gödel carried his analogy to its inevitable, momentous conclusion, which was to spell out for his readers (not symbol by symbol, of course, but via a precise set of “assembly instructions”) an astronomically long formula of PM that made the seemingly innocent assertion, “A certain integer g is not a prim number.” However, that “certain integer g ” about which this formula spoke happened, by a most unaccidental (some might say diabolical) coincidence, to be the number associated with (i.e., coding for) this very formula (and so it was necessarily a gargantuan integer). As we are about to see, Gödel’s odd formula can be interpreted on two different levels, and it has two very different meanings, depending on how one interprets it.

On its more straightforward level, Gödel’s formula merely asserts that this gargantuan integer g lacks the number-theoretical property called primness. This claim is very similar to the assertion “72900 is not a prime number”, although, to be sure, g is a lot larger than 72900, and primness is a far pricklier property than is primeness. However, since primness was defined by Gödel in such a way that it numerically mirrored the provability of strings via the rules of the PM system, the formula also claims:

The formula that happens to have the code number g

is not provable via the rules of Principia Mathematica.

Now as I already said, the formula that “just happens” to have the code number g is the formula making the above claim. In short, Gödel’s formula is making a claim about itself — namely, the following claim:

This very formula is not provable via the rules of PM.

Sometimes this second phraseology is pointedly rendered as “I am not a theorem” or, even more tersely, as

I am unprovable

(where “in the PM system” is tacitly understood).