▀ The Book of Sand of Borges and the Continuum of Cantor

The previous article

In the previous article entitled Nobody understands the infinite better than Borges I began the presentation of the short story The Book of Sand of the authorship of Jorge Luis Borges. In the story, The Book of Sand is a book that Borges acquired from a seller of books and Bibles that has the particularity that each time you open the book in one page, the page number changes, and the next page number is also changed. The book never had a first and last defined pages, and as if that were not enough, the text and illustrations never appear twice in the same place or on the same page.

The article ended like this:
Imagine that you open that "devilish book" and by some unknown power you can write the sequence of the page numbering as it appears page by page. Now close the book and reopen it again and repeat the process again. You will "finally" obtain all possible orderings of the natural numbers. I cannot show it here, but is possible to prove that your "list" of all possible orderings of the natural numbers is not countable, not even infinitely countable. Thus opening an closing The Book of Sand is an act of delving into the Continuum, an action that possibly Cantor never though of.
In the present article, which we can consider as a continuation of the one above, we will delve into an informal mathematical proof to show that the Book of Sand of Borges is much more than just an infinite book: it is a Transfinite book.

The infinitudes of Cantor

George Cantor conducted an extensive and deep research on the categories of the infinitude; something that was new to the mathematicians of his time. Prior to him it was assumed that all infinitudes were equal.

If we start from the series of the natural numbers 1, 2, 3, ... we say they are infinite; in fact, this series is the infinite series for excellence for its simplicity. But the even numbers series 2, 4, 6, ... it is also infinite, even when they seem to represent the "half" of the natural numbers. Another infinite set of integers is the set of prime numbers, despite the fact that as they progress the "distance" between them is widening. These sets were begun to be called "countably infinite" not because they were exactly "countable" but because they can be related in a one-to-one (1-1) relationship with the natural numbers we use to count.

More surprising is the fact that it can be shown that the fractions are also infinitely countable. Unexpectedly, with the proper arrangement of the elements, all fractions, such as 1/2, 4 /5, 458/245, ... can be infinitely listed, or counted as the first fraction, the second fraction, the third fraction, and so on.

Findings like these led to Cantor to deepen into the concept of "infinity" as we use it daily. Under a lot of opposition and humiliation, Cantor managed to formalize and establish the Theory of Sets, and the arithmetic of the infinites as a strong and indispensable field in the science of mathematics.

One of his sensational findings was demonstrating that there are infinite sets that are higher than the infinite set of natural numbers. There is no way to establish a one-to-one (1-1) correspondence between the natural numbers with those infinites, therefore, we must invent new categories for certain sets. These sets were called by Cantor Transfinite; infinites way beyond the infinity of the natural numbers.

The simplest example of transfinite sets are the set of the real numbers. Real numbers comprise those we commonly call decimal numbers and the infinite decimals that we never can end writing because they are
  • either "irrationals" as the square root of 2, commonly symbolized as √(2) = 1.414213562373...,
  • or they are transcendental numbers like the π =3.141592653589...
The discovery was unforeseen because it was not expected that the amount of irrational numbers was "so big" as to need to coin a new term and concept to study them. Note the reader that the contribution of the finite decimal numbers is almost null because the finite decimals can be expressed as fractions and we already mentioned that the fractions are "countable". For example, since 0.500... = 1/2, 0.333... = 1/3, etc. we can take away all those decimals out of the real line and still the remaining reals are transfinite.

This new set of numbers that can not be "counted" with the set of all natural numbers was the one to be known as The Continuum.

The properties of the Continuum are incredible: in the same way as the set of all even numbers is infinite, despite being a subset of the natural numbers, there are also ways to create subsets of real numbers that are equally transfinite as the set of all the real numbers itself. For example, the "small" line segment between the decimal 0.1 and the decimal 1.0 contains a transfinite number of real numbers. To this segment, later, we will give a good use with the Book of Sand of Borges.

The Book of Sand is a Transfinite book

After this brief digression of going deeper into what are the real numbers, in view of the fact that we will need them later, let us now return to the Book of Sand and make a "thought experiment" with it. Recall that the main peculiarity of this enigmatic and esoteric book was that the pages were randomly numerated and that it lacked a fixed first and last page because there were pages popping out of the nothingness at the beginning and at the end of the book.

Now imagine that you open this "devilish book" —as the Bible seller told Borges— and that for some unknown power you are able to write the sequence of the page numbers as they appear page by page. Now close the book and open it again and repeat the same process continuously. If you repeat this without stopping you will "finally" get all the possible combinations of sequences of pages of The Book of Sand.

What relationship exists between all the possible sequences of The Book of Sand of Borges and The Continuum of Cantor? Well, let's see if it is possible to establish unambiguous correspondences between all combinations of pages of The Book of Sand and The Continuum of Cantor.

To show the existence of unique relationships between the two sets requires a demonstration in two separate parts:
  1. That we can produce a function such that for every "book of sand" the function can assign an unequivocal decimal point in the Continuum. This is the relation A in the illustration.
  2. That there is also another function such that for every decimal in the Continuum the function can assign a unequivocal “book of sand”. That is the relation B in the illustration.
Will it be possible to demonstrate the existence of such mappings for the relations A and B as shown in the diagram?

This will be seen in the next post.

Articles in this series
1 - ▀ Nobody understands the infinite so well as Borges
2 - ▀ The Book of Sand of Borges and the Continuum of Cantor
3 - ▀ The Book of Sand is a Transfinite Book
4 - ▀ Is The Book of Sand a book from the fourth dimension?