I Am a Strange Loop

Fortunately, there do exist strange loops that are not illusions. I say “fortunately” because the thesis of this book is that we ourselves — not our bodies, but our selves — are strange loops, and so if all strange loops were illusions, then we would all be illusions, and that would be a great shame. So it’s fortunate that some strange loops exist in the real world.

On the other hand, it is not a piece of cake to exhibit one for all to see. Strange loops are shy creatures, and they tend to avoid the light of day. The quintessential example of this phenomenon, in fact, was only discovered in 1930 by Kurt Gödel, and he found it lurking in, of all places, the gloomy, austere, supposedly paradox-proof castle of Bertrand Russell’s theory of types.

What was a 24-year-old Austrian logician doing, snooping about in this harsh and forbidding British citadel? He was fascinated by paradoxes, and although he knew they had supposedly been driven out by Russell and Whitehead, he nonetheless intuited that there was something in the extremely rich and flexible nature of numbers that had a propensity to let paradox bloom even in the most arid-seeming of deserts or the most sterilized of granite palaces. Gödel’s suspicions had been aroused by a recent plethora of paradoxes dealing with numbers in curious new ways, and he felt convinced that there was something profound about these tricky games, even though some people claimed to have ways of defusing them.

Mr Berry of the Bodleian

One of these quirky paradoxes had been concocted by an Oxford librarian named G. G. Berry in 1904, two years before Gödel was born. Berry was intrigued by the subtle possibilities for describing numbers in words. He noticed that if you look hard enough, you can find a quite concise description of just about any integer you name. For instance, the integer 12 takes only one syllable to name, the integer 153 is pinpointable in but four syllables (“twelve squared plus nine” or “nine seventeens”), the integer 1,000,011 is nameable in just six syllables (“one million eleven”), and so forth. In how few syllables can you describe the number 1737?

In general, one would think that the larger the number, the longer any description of it would have to be, but it all depends on how easily the number is expressible in terms of “landmark” integers — those rare integers that have exceptionally short names or descriptions, such as ten to the trillion, with its extremely economic five-syllable description. Most large numbers, of course, are neither landmarks nor anywhere near one. Indeed, by far most numbers are “obscure”, admitting only of very long and complex descriptions because, well, they are just “hard to describe”, like remote outposts located way out in the boondocks, and which one can reach only by taking a long series of tiny side roads that get ever narrower and bumpier as one draws nearer to the destination.

Consider 777,777, whose standard English name, “seven hundred seventy-seven thousand seven hundred seventy-seven”, is pretty long — 20 syllables, in fact. But this number has a somewhat shorter description: “777 times 1001” (“seven hundred seventy-seven times one thousand and one”), which is just 15 syllables long. Quite a savings! And we can compress it yet further: “three to the sixth plus forty-eight, all times ten cubed plus one” or even the stark “the number whose numeral is six sevens in a row”. Either way, we’re down to 14 syllables.

Working hard, we could come up with scads of English-language expressions that designate the value 777,777, and some of them, when spoken aloud, might contain very few syllables. How about “7007 times 111” (“seven thousand seven times one hundred eleven”), for instance? Down to 13 syllables! And how about “nine cubed plus forty-eight, all times ten cubed plus one”? Down to 12! And what about “thrice thirty-nine times seven thousand seven”? Down to just 11! Just how far down can we squeeze our descriptions of this number? It’s not in the least obvious, because 777,777 just might have some subtle arithmetical property allowing it to be very concisely expressed. Such a description might even involve references to landmark numbers much larger than 777,777 itself!

Librarian Berry, after ruminating about the subtle nature of the search for ever shorter descriptions, came up with a devilish characterization of a very special number, which I’ll dub b in his honor: b is the smallest integer whose English-language descriptions always use at least thirty syllables. In other words, b has no precise characterization shorter than thirty syllables. Since it always takes such a large number of syllables to describe it, we know that b must be a huge integer. Just how big, roughly, would b be?