by Chris Bodenner
Wednesday would have been Jorge Luis Borges' 112th birthday, so I thought this quote was rather pertinent to the topic at hand: "One concept corrupts and confuses the others. I am not speaking of the Evil whose limited sphere is ethics; I am speaking of the infinite."
I find it impossible to believe there has been so much high-level discussion of infinity and not yet one mention of Jorge Luis Borges, the poet laureate of infinity, whose birthday was just commemorated on Google's homepage. Borges has several stories dealing with infinity, perhaps most obviously "The Aleph", and in a subtler way, "The Blue Tigers". But his most famous story is a direct meditation on infinity: "The Library of Babel". I remember reading that story and my mind recoiling in shock at the very existence of the notion of infinity.
In the story, there is a supposedly infinite library of infinite texts. Since the texts are infinite, every permutation must appear, which means there must be one text which is a perfect index of the others. But the narrator tells us that fanatics went through the library, looking for this index, trying to burn it. Thus, it might be lost. But that's okay, our narrator says. Since there are infinite texts, there must be an infinite set of texts which match the index with the exception of a single character; and these would be as good, for all intents and purposes, as the "true" index.
If anyone finds your infinity series fascinating and has not read Borges, I would recommend they do so with all due haste.
I omitted another fun fact about infinities. This particular one is most appropriate on the birthday of Jorge Luis Borges, writer of The Library of Babel. This is a beautiful and eerie surrealist story which is largely about man's struggle to comprehend and draw meaning from the infinite, and I can't recommend it (or his other work) enough.
Hopefully you recall something of the difference between countably and uncountably infinite sets, and recall that the real numbers are among the latter. Do you know what else is only countably infinite? The number of possible English words, sentences, and books. (It's easy to enumerate them, numbering as you go: "A, B, C, … Z, AA, AB, AC,…,AZ, BA, BB,…,ZZ,…,AAA…etc." Throw in a few punctuation marks and you're basically there.)
Same for mathematical equations and other expressions. There aren't that many symbols and ways to arrange them, and even the craziest (compound fractions, etc.) can be broken down into a sequence of consecutive symbols on one line if we want to do so. And sure, you can always define new symbols or mathematical terms, but how do you do that? You write a book about it! And there are only countably many books you can write, whether they contain equations or not! Even if you're willing to use mathematical terms and symbols loosely and require the reader to fill in the gaps (for instance, the use of "…" in the previous paragraph), it doesn't solve the root problem.
What's this mean for us? Using the real numbers as our example again, there are some real numbers that we'll never be able to calculate, define, or even describe in any manner that would identify them uniquely. A thousand monkeys with a thousand typewriters could write Shakespeare; they could also acquaint us with "The square root of 2" or "Pi: The ratio between the circumference and diameter of any circle." But unfathomable infinities of numbers lie forever beyond their reach and ours.
I'm not really the sort of guy who has a favorite number; I'm more awed by the interconnection between different elements in a mathematical system than any individual one. Still, I can't help but be a little sad at all the numbers, many of them probably with fascinating properties, that are totally beyond our ability to conceptualize. And of course even if you don't care about numbers, or about real numbers specifically, it's worth remembering there are sets of objects that make the reals look downright tiny! Infinity is a big place.
(Image of a "Library Of Babel" etching by Eric Desmazieres. More versions here.)