The Infinite Library
Arguably the strangest part of mathematics is the consideration of infinity. There is a kind of mysticism concerning limits at infinity that is left unadressed in an introductory calculus class where infinity is taken more as a tool than a mathematical object.
One of the most intriguing expansions on the concept of infinite is the differentiation of higher infinities. At first, this doesn’t seem like it should be possible, after all what is larger than infinity? The answer to that question was provided by the ever insightful Georg Cantor who first pioneered the idea. One of the ways his idea concerning uncountability can be realized is the infinite library.
The Infinite Library
Consider a library of vast size. It is filled with shelves each packed with books, each seemingly identical a hundred pages long. You walk in and open the first book on the first shelf only to find it blank. On the second you find printed a single a
. On the third aa
, on the fourth, aaa
. Eventaully the entire alphabet
abcdefghijklmnopqrstuvwxyz1234567890,?!'":;()/.
is enumerated through in every possible combination that can be fit within the 1000 page book. Within this library there are more books than there are atoms in the universe. Over \(7.9 \times 10^{262}\) volumes, that is
79766443076872509863361000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
books. It’s an unfathomable number, made more so by what this library would contain. Within it you would find the first 100 pages of every work of shakespear. You would find unpublished volumes never written. Within it would be definitive proof or counterexample to the Riemann Hypothesis. All of human knowledge that ever was or ever will be expressed within a hundred pages would be at home in the library.
Now imagine that each book within the library were infinite, wouldn’t the library containing all such infinite books be something even larger? It turns out we can show this.
The Math
The standard form of infinity we are used to is called “countably” infinite. It’s analog is an infinite book, where ever character can be put in a line and enumerated (in mathematical terms any set which can be surjectivly projected onto by the natural numbers). Does this library conform to this same notion of infinity?
Suppose the books in the library could be enumerated as \(B_1, B_2, B_3, \dots\) and consider each book as an infinite sequence of characters, i.e
\[B_i = ldsahfkjasdfbvkjbadcjkladfbgjkldafbgkfdhagkljdfkjlbadkfb...\]
Were the library countable than every book within it could be enumerated in the list. Our goal then, is to construct a book within the library that is outside of any enumeration. This is commonly known as Cantor’s Diagonalization Argument and the origin of the notion of “uncountable.”
Suppose we consider the list of books as
\[ \begin{align*} B_1 &= asdfbadf\dots \\ B_2 &= lakjhdsf\dots \\ B_3 &= ptsadfhg\dots \\ B_4 &= hafgadjl\dots \\ &\vdots \end{align*} \]
Then let us construct a new book, \(B'\) as follows. Let the \(i\)th character of \(B'\) be the character following the \(i\)th character of \(B_i\). That is, supose the 2nd character of \(B_2\) were \(a\), then the second character of \(B'\) would be \(b\), and so on. (Perhaps \(z\) would map to \(a\) or \(.\) to \(a\) if we allow special characters in our book, then the cycle would repeat). In the case above we would have
\[ B' = bbth... \]
It’s easy to see that for every \(i\), the \(i\)th character of \(B_i\) will be distinct from the \(i\)th character of \(B'\) by our construction. But this is exactly the book we were looking for! Since \(B'\) is distinct from every \(B_i\) it clearly isn’t an element of the infinite sequence \(B_1, B_2, B_3, \dots\) but is just as clearly a book in its own right. Hence, the library we have described can’t be countable and we must seek out the new world uncountable.
Consequences
The above diagonalization argument works for many sets, notably it is commonly used to show that the real numbers are uncountable. Indeed, replace our books with the decimal representations of real numbers and you get exactly the same proof. However, it leads to further thoughts on infinity. Is there something greater than uncountable? Well the same argument can be applied to ever greater sets, building up an (ironically countable) set of infinities. Going further there are even greater levels of infinity beyond the countable iterations, and it’s a fascinating topic that kept Georg Cantor up for much of his life.
However, as soon as we hit uncountable things start to get interesting. Much of the most interesting examples in calculus such as the Cantor set or the Devil’s staircase arise naturally as a consequence of the interplay between countable and uncountable sets.