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November 23, 2007

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Ben, your philistinism will not avail you against Borges. For there are clearly non-standard books in the Library. For one, the library contains "books in the Crimson Hexagon: books whose format is smaller than usual, all-powerful, illustrated and magical."

Moreover, perhaps the Library is understood to extend through space as well as time, in which case any possible non-repetitive multivolume sets may be brought into existence simply by rearranging the order of books, through human agency. And then the Library is truly complete over its entire time line.

Selah.

Philistinism? Surely this is just the sort of attention that Borges would have appreciated.

Saying that overlapping sets of nonrepetitive multivolume texts can coexist because you can have first one at time t0, then the other at time t1, and the texts coexisted in the range (t0,t1) is such blatant sophistry I'm not surprised it came from a political scientist.

The existence of nonstandard sets is only tolerable so long as they are relatively few in number, on pain of making the whole thing too silly to bear (or undermining the interest of the story—you may say either). Hence Borges' wise limitation of such things to particular hexaga.

But the library is ever-changing! ("the senseless perdition of millions of books") Books are being destroyed, and, perhaps books are even being created anew? All claims about the vast extent of what the library "contains" become trivially false if we limit the scope of the claim to one point in space-time. Piffle! (And of course the library, being the universe, doesn't have a discrete point in time at which it began, when it could be said to be complete.)

Relatively few in number! But the Library is (possibly) infinite. A billion hexaga would be relatively few... perhaps even a countably infinite hegaga... I feel an invocation of Cantor coming on...

The library can't be infinite, even if the narrator calls it "perhaps infinite", because "the fundamental law of the library" is that "There are not, in the whole vast Library, two identical books" (italics in original). Given the composition of the books, this already allows us to put an upper bound on the number of books: 25**s + C, C being a comparatively small number equal to the number of books in the Crimson Hexagon and what similar other hexaga there might be. It really does have to be comparatively small. Take a moment to convince yourself of this. Note also that it's far from clear that there is such a hexagon in the library; the narrator says that those who destroyed some of the books were "spurred by the delirium of storming the books in the Crimson Hexagon". There is no direct endorsement of the existence of such a place. Upper bound because, as you correctly point out, some books have been destroyed. And in this same place the narrator says that "the library is so enormous [sc. it is not infinite] that any reduction undertaken by humans is infinitesimal" (emph. added), and that "each book is unique, irreplaceable" (emph. added).

The uniqueness constraint is all that's necessary to establish the incompleteness of the library as described in the post, anyway.

I think you misspelled Borges' name.

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