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Look out, it’s evil!
Helping to drag America, kicking & screaming, into the Age of Enlightenment
6_bleen_7
6_bleen_7
Self-cataloguing Sentences
Consider the following:

L is the first and twenty-sixth letter in this sentence.

It's a simple modification of the following infinitely long sentence discovered by Dr. J. K Aronson, and quoted (not the whole thing, obviously) in Douglas Hofstadter's Metamagical Themas:

T is the first, fourth, eleventh, sixteenth, twenty-fourth, twenty-ninth, thirty-third, ...

Dr. Aronson's sentence is infinitely more clever than mine, as well as infinitely longer—he had the original idea, after all—but mine has one advantage: It's a non-trivial variation on the theme. See, using the letter T guarantees an infinite progression, because nearly all ordinals contain at least one T; the first two exceptions are second and one hundred second. Replacing the T with an H also virtually ensures a never-ending sentence, though fewer ordinals have an H. At the other end of the scale, any letter in the set {ABDGJKMOPQUVWXYZ}, when substituted for the asterisk, yields the equally trivial

* is the first letter in this sentence.

The interesting cases are the letters, like my L, that don't nucleate either an infinite or a "first letter only" type of sentence. And the longer they are, the more intriguing, because of the difficulty in finding a solution. For example, does this sentence end?

I is the first, second, eighth, nineteenth, twenty-fifth, forty-first, fifty-first, fifty-sixth, sixty-first, sixty-sixth, seventy-first, seventy-sixth, eighty-first, eighty-sixth, ninety-first, one hundred third, one hundred fifteenth, one hundred twentieth, one hundred twenty-sixth, one hundred thirty-first, ...(.?)

I suspect it doesn't, but I'm not going to attempt a mathematical proof.

It's also possible that for some letters there is no solution. I tried starting one with N, but I got caught in a loop, in which adding a number for the third-to-last N added two Ns before it and therefore required inserting two ordinals before the one I'd just put in. Those earlier numbers moved the last three words and thus made superfluous the one I'd added previously. But if I took them back out, I needed them again, &c., &c.



Wouldn't it be neat if we could write a sentence that listed all its contents, instead of just the instances of one letter? How about this one (by Lee Sallows, again, from Metamagical Themas)?
This pangram tallies five As, one B, one C, two Ds, twenty-eight Es, eight Fs, six Gs, eight Hs, thirteen Is, one J, one K, three Ls, two Ms, eighteen Ns, fifteen Os, two Ps, one Q, seven Rs, twenty-five Ss, twenty-two Ts, four Us, four Vs, nine Ws, two Xs, four Ys, and one Z.
It was generated by computer, using an exhaustive search algorithm. (A pangram is a sentence that contains all twenty-six letters of the alphabet.) Sallows earlier composed a more comprehensive self-cataloguing sentence; though despite its more thorough completeness it is not as impressive, as the beginning section is expanded and clearly contrived to make the numbers work out. Also, it contains no Js, Qs or Zs. (The pangram above is just about as terse as can be—it approaches perfection. He first tried to generate the Platonic ideal, which begins "This pangram contains...", but failed.)
Only the fool would take trouble to verify that his sentence was composed of ten a's, three b's, four c's, four d's, forty-six e's, sixteen f's, four g's, thirteen h's, fifteen i's, two k's, nine l's, four m's, twenty-five n's, twenty-four o's, five p's, sixteen r's, forty-one s's, thirty-seven t's, ten u's, eight v's, eight w's, four x's, eleven y's, twenty-seven commas, twenty-three apostrophes, seven hyphens, and, last but not least, a single !

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Comments
samwibatt From: samwibatt Date: August 19th, 2006 02:51 pm (UTC) (Link)
This looks like a fun thing to write programs about. If you had an ordinal generator, it should be a snip - might be able to check the "I" sentence for conversion.

Neato! Been awhile since I've read Metamagical Themas. Do you have the copy I tore the cover off of while ngo?
6_bleen_7 From: 6_bleen_7 Date: August 19th, 2006 05:09 pm (UTC) (Link)
Ngo—I mean no; I bought a cover-intact copy after we moved to Seattle.

Heh-heh—I was thinking about that whilst writing the entry; and also, "How long does it have to stay there?"
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