Infinite extent
In mainstream theories of natural language syntax, every syntactically valid utterance can be extended to produce a new, longer one, because of recursion.[3] However, the books in the Library of Babel are of bounded length ("each book is of four hundred and ten pages; each page, of forty lines, each line, of some eighty letters"), so the Library can only contain a finite number of distinct strings. Borges' narrator notes this fact, but believes that the Library is nevertheless infinite; he speculates that it repeats itself periodically, giving an eventual "order" to the "disorder" of the seemingly random arrangement of books. Mathematics professor William Goldbloom Bloch confirms the narrator's intuition, deducing in his popular mathematics book The Unimaginable Mathematics of Borges' Library of Babel that the library's structure necessarily has at least one room whose shelves are not full (because the number of books per room does not divide the total number of books evenly), and the rooms on each floor of the library must either be connected into a single Hamiltonian cycle, or possibly be disconnected into subsets that cannot reach each other.[4]
Quine's reduction
W. V. O. Quine notes that the Library of Babel is finite, and that any text that does not fit in a single book can be reconstructed by finding a second book with the continuation. The size of the alphabet can be reduced by using Morse code even though it makes the books more verbose; the size of the books can also be reduced by splitting each into multiple volumes and discarding the duplicates. Writes Quine, "The ultimate absurdity is now staring us in the face: a universal library of two volumes, one containing a single dot and the other a dash. Persistent repetition and alternation of the two are sufficient, we well know, for spelling out any and every truth. The miracle of the finite but universal library is a mere inflation of the miracle of binary notation: everything worth saying, and everything else as well, can be said with two characters."[5]
