I Am a Strange Loop

Any large number that you run into in a newspaper or magazine or an astronomy or physics text is almost surely describable in a dozen syllables, twenty at most. For instance, Avogadro’s number (6×10²³) can be specified in a very compact fashion (“six times ten to the twenty-third” — a mere eight syllables). You will not have an easy time finding a number so huge that no matter how you describe it, at least thirty syllables are involved.

In any case, Berry’s b is, by definition, the very first integer that can’t be boiled down to below thirty syllables of our fair tongue. It is, I repeat, using italics for emphasis, the smallest integer whose English-language descriptions always use at least thirty syllables. But wait a moment! How many syllables does my italicized phrase contain? Count them — 24. We somehow described b in fewer syllables than its definition allows. In fact, the italicized phrase does not merely describe b “somehow”; it is b’s very definition! So the concept of b is nastily self-undermining. Something very strange is going on.

I Can’t Tell You How Indescribably Nondescript It Was!

It happens that there are a few common words and phrases in English that have a similarly flavored self-undermining quality. Take the adjective “nondescript”, for instance. If I say, “Their house is so nondescript”, you will certainly get some sort of visual image from my phrase — even though (or rather, precisely because) my adjective suggests that no description quite fits it. It’s even weirder to say “The truck’s tires were indescribably huge” or “I just can’t tell you how much I appreciate your kindness.” The self-undermining quality is oddly crucial to the communication.

There is also a kind of “junior version” of Berry’s paradox that was invented a few decades after it, and which runs like this. Some integers are interesting. 0 is interesting because 0 times any number gives 0. 1 is interesting because 1 times any number leaves that number unchanged. 2 is interesting because it is the smallest even number, and 3 is interesting because it is the number of sides of the simplest two-dimensional polygon (a triangle). 4 is interesting because it is the first composite number. 5 is interesting because (among many other things) it is the number of regular polyhedra in three dimensions. 6 is interesting because it is three factorial (3×2×1) and also the triangular number of three (3+2+1). I could go on with this enumeration, but you get the point. The question is, when do we run into the first uninteresting number? Perhaps it is 62? Or 1729? Well, no matter what it is, that is certainly an interesting property for a number to have! So 62 (or whatever your candidate number might have been) turns out to be interesting, after all — interesting because it is uninteresting. And thus the idea of “the smallest uninteresting integer” backfires on itself in a manner clearly echoing the backfiring of Berry’s definition of b.

This is the kind of twisting-back of language that turned Bertrand Russell’s sensitive stomach, as we well know, and yet, to his credit, it was none other than B. Russell who first publicized G. G. Berry’s paradoxical number b. In his article about it in 1906, Gödel’s birthyear (four syllables!), Russell did his best to deflect the paradox’s sting by claiming that it was an illusion arising from a naïve misuse of the word “describable” in the context of mathematics. That notion, claimed Russell, had to be parceled out into an infinite hierarchy of different types of describability — descriptions at level 0, which could refer only to notions of pure arithmetic; descriptions at level 1, which could use arithmetic but could also refer to descriptions at level 0; descriptions at level 2, which could refer to arithmetic and also to descriptions at levels 0 and 1; and so forth and so on. And so the idea of “describability” without restriction to some specific hierarchical level was a chimera, declared Russell, believing he had discovered a profound new truth. And with this brand-new type of theory (the brand-new theory of types), he claimed to have immunized the precious, delicate world of rigorous reasoning against the ugly, stomach-turning plague of Berry-Berry.

Blurriness Buries Berry