In my preceding article
The Book of Sand of Borges and the Continuum of Cantor I wrote about the manifest similarities of the appreciations of the concept of infinitude between the literature writer and Argentinian academic Jorge Luis Borges and the German mathematician George Cantor. The article was at the same time a continuation of
Nobody understands the infinite so well as Borges, which is the article number one in this series.
To understand the three articles the reader is encouraged to be related with Borges’ short story
The Book of Sand, or read the articles in sequence.
Is The Book of Sand exactly one book, or a book that reshuffles itself every time it is opened?
All the discussion that follows from here to the end of the current article relies on a personal interpretation of the short story
The Book of Sand. For me, the book renovates itself every time somebody opens it. Let me quote again the segments of the story that leads to my interpretation. First Borges finds an illustration in the book:
...I turned the leaf; it was numbered with eight digits. It also bore a small illustration, like the kind used in dictionaries —an anchor drawn with pen and ink...
Then the stranger warns him about the infinitude of the book:
It was at this point that the stranger said: "Look at the illustration closely. You'll never see it again".
Borges challenges the vendor by marking the illustration and tries to find it again:
I noted my place and closed the book. At once, I reopened it. Page by page, in vain, I looked for the illustration of the anchor...
the illustration disappeared, or at least he didn't find it. Based on this words is that in my interpretation, the book Borges that bought was not unique in its composition of pages: it was a book that in some mysterious way regenerated itself every time it was reopened. I call every reopening an
instance of the book, so each instance is another "book of sand". That explains why in one instance he sees an illustration, and in another instance (another reopening, another self-reshuffling) the illustration was not in its previous place. That is, the illustration of the anchor belonged to an ephemeral instance of all the possible instances of the infinite and incredible
Book of Sand.
So for me, the book was infinite in pages and at the same time was an infinitude of books all of them packed into a single one. That multiplicity of infinitudes is the ground upon which we will build and prove the assertion that the
Book of Sand is something more than an infinite book: it is a
transfinite book.
Relations between two infinite sets
I suggested in the preceding article that Borges’ imaginary book is more than an infinite book:
The Book of Sand is a transfinite book.
Transfiniteness —a concept introduced to modern math by Cantor— was briefly described in that article. It was also stated in this article that in order to prove that
The Book of Sand is a
Transfinite book, we must find a function that could establish correspondence between all the possible "books of sand" and the
Continuum of the real numbers, and another function that could establish unique correspondence between elements of the Continuum and a corresponding unique "books of sand".
The following drawing, taken from the previous article, will be reused to refresh our quest for mapping between two sets of objects. We will focus on the possible existence of two relations
A and
B between the two infinite collections of our study. We are heading to prove that there exist such
A and
B relations, although they need not be necessarily reciprocals.
The point here is the following: we can establish a two-way (1-1) correspondence between every natural number and every even number: simply double every natural number and we'll have even numbers, or simply half any even number we'll obtain a natural number. In this simple example, the relation of doubling and the relation of halving are mutually reciprocal. In this example, the relation of doubling is (1-1) and the relation of halving is also (1-1).
Let us now take the squaring relation for the natural numbers; in this case for each natural number
x there exist one and only one number
x2. The number 2 when squared produces the number 4, and the number -2 when squared also produces the number 4, but note that squaring never produces two different results at the same time. On the other hand, the “reciprocal” function of the squaring function, the square root function, is not (1-1). The function f(
x) =
x2 is a map that assigns to any number
x its unique square
x2, but its inverse function, the one that assigns the square root to a natural number is not unique, because, as we have seen, for example, 4 has two the square roots: +2, and –2; therefore the square root mapping is not unique. To accomplish some (1-1) mappings we must split this function in two separate (1-1) maps:
g(
x) = +√(
x) and
h(
x) = –√(
x).
So the noteworthy fact about mappings, relations, and functions (they are loosely synonymous) is that they produce unique results. What we want is, for example, that if are to compute the area of a square or a rectangle, the result is one and only one unique number. It doesn't matter we are working with finite or infinite sets of objects or numbers; the mapping rules that apply are the always the same: the uniqueness of the result.
With this ultra-brief introduction to mappings, let us begin our task of mapping the set of all possible "books of sand" to the set of all decimals in the number line. To achieve this goal we will divide our arguments into two parts:
- Part A is dedicated to proving that there exists an A-relation, where to every "book of sand" we can associate a unique decimal of a chosen subset of the real numbers, and
- Part B is dedicated to proving that there exists a B-relation, where to every decimal of a chosen subset of the real numbers we can associate a unique "book of sand".
Part A. For every instance of the Book of Sand there is a unique decimal in the Continuum
To Cantor and his
Theory of Sets, the
Continuum is the dense and compact set of all real numbers. But there are also many instances in which
Continuum are also some special subsets of all decimals such as the decimals between 0.1 and 1.0. In this "little" interval of decimals are the real numbers: 0.1297067, 0.96239..., and so on. In passing we must note that the number 1.0 and the decimal 0.999 ... are equivalent: 1.0 is just another way to write the decimal 0.999 ... This fact is simply demonstrated by adding 1.0 plus 0.999... and dividing by 2 to find which number is between both numbers. Hence, it is of enormous help to work with the
Continuum between 0.1 and 1.0 instead of working with all the real numbers together.
To explore the relationship between the
Book of Sand Borges and the
Continuum of Cantor suppose that we label by italicized letters as for example
s1,
s2, ... each sequence numbering of our "thought experiment" (see the previous article) suggested above. So, among the possibilities of series of pages we have,
s1 = {... 40514, 999, ... } (The example of pages given by Borges)
s2 = {... 280, 45, ... } (Any other instance of the book)
Let’s join together all the digits of the pages to make infinitely large numbers, as follows:
n1 = 40514999 ...
n2 = 28045 ...
Now we turn those integers into decimals and label the possible decimals by
d1,
d2, ... by simply adding a decimal point to the left, like this:
d1 = 0.40514999 ...
d1 = 0.28045 ...
So to the sequences of pages
s1 of the pages ..., 40514, 999, ... we are assigning the decimal
d1 = 0.40514999 ..., and to the sequence
s2 of the pages ..., 280, 45, ... corresponds the decimal
d2 = 0.28045 ...
Have we demonstrated that there is a unique relationship
A from the set of all "books of sands" to the decimals as we specified in the diagram? Can we apply a label decimal between 0.1 and 1.0 for every possible sequence numbering of the pages of the
Book of Sand by just following the rules specified above?
Unfortunately, no. No matter how convincing the rules may look, those rules are not enough to produce unique mappings. To see why, consider the following different set of pages
t1 = {... 40, 51, 4999, ... } (Any other instance of the book)
t2 = {... 2804, 5, ... } (Any other instance of the book).
Note that under the rules above they also produce the same results
g1 = 0.40514999 ...
g2 = 0.28045 ...
That is,
s1 ≠
t1, but
d1 =
g1. Similarly,
s2 ≠
t2 but
d2 =
g2. Clearly, the rules we are attempting do not produce or generate unique results.
We must seek, then, a special rule to give a series of pages really unique decimal numbers only.
Two important properties of the prime numbers
We can use two fundamental properties of the
prime numbers to construct the mapping we need to associate to each
Book of Sand a unique decimal. The properties are:
1. For each natural number, there is exactly one unique prime number.
2. Each composite natural number can only be decomposed in only one set of prime factors.
What states property number one is that the prime numbers are infinite. As a short review of the prime numbers, the first prime is 2, the second prime is 3, the prime number 20 is 71 and so on. We denote the series of prime numbers by lowercase letter
p’s as follows:
p1 = 2,
p2 = 3, ...,
p20 = 71, ...
When we do not have a Table of Primes at hand, one good resource for finding the
n-th prime number is
The Nth Prime Page. A prime page by Andrew Booker. For the actual computations below this online server was used.
What states property number two is that the numbers that are not primes, like 4, 6, 9, etc. can only be factored in unique sets of prime numbers; for example, the number 220 = 2 × 2 × 5 × 11, but there is no other way to factor 220, there are no other prime numbers which multiplied give the same result 220.
Using the prime numbers
Returning to the example of the page numbers in the first sequence
s1 let us multiply the prime number
in the position 40514 by the prime number
in the position 999, and so on. Since now we are dealing with a new map, then we also obtain new numbers
n1 and
n2, as follows:
n1 =
p40541 ×
p999 × ... for the sequence
s1
and for the sequence
s2
n2 =
p280 ×
p45 × ....
Now, by property number two of the prime numbers, both numbers
n1 and
n2 are unique and different from each other. That is, under this new assignment, different page sequences generate different prime numbers multiplications.
The steps we now take are directed to obtain unique decimals from those unique prime numbers multiplications.
Since
p40541 = 487183 and
p999 = 9707 then
n1 = 487183 × 9707 × ... i.e.
n1 = 4729085381 × ...
On the other hand, for the number
n2, we have
n2 = 1811 × 197 × ... = 356767 × ...
given that
p280 = 1811 and
p45 = 197.
But what we’ll have for the sequences t
1 and t
2 in the examples above? Well, since t
1 = {... 40, 51, 4999, ... } and t
2 = {... 2804, 5, ... } etc., then
N1 =
p40 ×
p51 ×
p4999 × ..., and
N2 =
p2804 ×
p5 × ...
Clearly,
n1 is not equal to
N1 and
n2 is not equal to
N2, even when the digits of the set
s1 are the same digits used in
t1, and the digits of the set
s2 are the same digits used in
t1. This is due to property 2 of the prime numbers above: the multiplication of different primes necessarily produce different results.
Continuing with this part of the proof, let us now denote the digits of the number
N1 by
d1,
d2, ... and the digits of the number
N2 by
D1,
D2, etc. Then
N1 =
d1 d2 d3 ... and
N2 =
D1 D2 D3 ...
As an example, for the random number 3735,
d1 = 2,
d2 = 7,
d3 = 3 and
d4 = 5.
Therefore, for the sets
s1 and
s2, we have
The sequence
s1 = {... 40514, 999, ... } can be uniquely mapped to the decimal 0. 587576...., and
the sequence
s2 = {... 280, 45, ... } can be uniquely mapped to the decimal 0.75476....
That is, we have managed to produce a mapping from every instance of the "book of sand" to a unique decimal in the interval of 0.1 to 1.0.
Wrapping up the rules of this mapping
How can we assign a unique decimal to a given
Book of Sand if there is never a first page, and consequently, never the first factor to find
N1 or
N2?
Borges mentions that the
Book of Sand had no first and last page, that’s the reason why we write each sequence of pages beginning and ending with ellipses. Then how can we deal with the product of the elements of a sequence of numbers that has no beginning and no end? There may exist a page with the number 1 on it, but that doesn’t necessarily constitute being its first page.
Since the multiplication of the numbers is commutative, the order in which we write the pages sequence is irrelevant for the final result. Therefore, we can equally write the elements omitting the first ellipsis as shown below
s1 = { ... 40514, 999, ... } = { 40514, 999, ... }, and
s2 = { ...280, 45, ... } = { 280, 45, ... }.
Therefore
s1 = {40514, 999, ... } -------> 0. 587576...., and
s2 = {280, 45, ... } -------> 0.75476....
In this way, we have managed to associate with each
Book of Sand a single and unique decimal in the segment of real numbers from 0.1 to 1.0.
Now we summarize in symbols the function we worked.
Let
Sb be any sequence of page numbers of some
Book of Sand with elements
e1,
e2,
e3 ... etc. that is,
Sb = {
e1,
e2,
e3 ... }. Note that this is equivalent to say that the page numbers are
e1,
e2,
e3 ...
Let
Nb be the product of all the primes with positions
pe1,
pe2,
pe3, etc. That is,
Nb =
pe1 ×
pe2 ×
pe3 × ...
Let
Db be the decimal obtained when we add a decimal point in front of
Nb. Suppose the decimal digits of
Db be the digits
d1,
d2,
d3, etc. Then, the final function
F that assigns a random
Book of Sand S
b to a decimal in the
Continuum is:
F (
Sb) = 0.
d1 d2 d3 ....
In mathematical terms, the set of all "books of sand" is called the
domain of the function
F, and the set of all decimals obtained under
F is called the
range of the function. In order for a function between a domain and a range to have mathematical significance, it can be one-to-one or many-to-one but not one-to-many.
Under the mapping we are dealing with, any sequence
Sb of the
Book of Sand can also be called a
pre-image or an element of the domain of all the possible copies of the
Book of Sand, and its corresponding decimal
Db is also called an
image.
Some thoughts about this mapping
Note that with this function we can assign any possible
Book of Sand to a unique decimal between 0.1 and 1.0. However, some decimals —in fact, infinitely many decimals— will remain without its copy of a
Book of Sand, as for example, the decimal 0.385. Why this? Is it permissible?
If every
Book of Sand has infinitely many pages, then under the function
F, any book
F (
Sb) must have an associated number
Nb =
p(e1) ×
p(e2) ×
p(e3) × ... of infinitely many prime factors. For this reason, any ending decimal will NEVER be an instance of the
Book of Sand.
However, generating
every decimal of the
Continuum between 0.1 and 1.0 is not a requisite for the function
F (
Sb) to be a valid mapping.
The requisite is that for every instance of a Book of Sand a different decimal be generated.
Part B. For every nonrational decimal between 0.1 and 1.0 there is a different Book of Sand
We have gone past the first part of the proof, now we must now prove Part
B of the chart, that is, that for every nonrational decimal between 0.1 and 1.0, there is a sequence of numbers of pages related in the
Book of Sand. Now we are going to demonstrate that forever
nonrepeating decimal in the range 0.1 to 1.0 we can define a function that assigns an instance, that is, a different and unique "book of sand" to that decimal.
Take any decimal between 0.1 and 1.0, for example, 0.52826971068507 ... Let us convert it into a whole number by eliminating the decimal point and then gradually subdivide its digits in groups of 1-digit, 2-digits, 3-digits, and so on like this: [5] [28] [269] [7106] [85907] ....
That to that this decimal 0.52826971068507 ... it corresponds an instance, a "book of sand" with pages 5, 28, 269, 7106, 85907, and so on.
But, what would happen if one or more of the “pages” begin with zero, for example, [5], [28], [028], [0005], ... Which instance would be the associated for the decimal 0.5280280005 ...?
This case warns us that we must improve the correspondence between the decimals and the instances of the
Book of Sand. One way to improve the mapping it is to add one or more digits corresponding to the place where they make the partition of pages.
This is the way it will work for the decimal in the example: 5 is the first partition, 28 the second, 028 the third, and so on. We will then have, that to the decimal 0.5280280005 ... now corresponds the book with pages [
15], [
228], [
3028], [
40005], ...
This is a solution that guarantees us that no page number begins with a zero.
Now we state in symbols how this map behaves.
Let
dc be a
nonrational decimal chosen between 0.1 and 1.0. Let
d1,
d2,
d3, ... be the digits of this decimal. Let
ni be natural numbers made by the digits of this decimal as follows:
n1 = 10
1 +
d1
n2 = 10
2 +
d2 × 10 +
d3
n3 = 10
3 +
d4 × 10
2 +
d5 × 10 +
d6
Therefore, to the decimal
dc made of the decimals
d1,
d2,
d3, ... corresponds the set made up of the above sequences:
Sn ={[
n1], [
n2], [
n3], ...} Each one of the numbers enclosed in brackets corresponds to a page numbered by the number within the brackets.
We can denote this mapping from the decimals to the instances of "the book of sand" by the symbol
G(
dc) = S
n.
Recalling the example above, to the decimal 0.5280280005 ... corresponds the book with pages [15], [228], [3028], [40005], ... This mapping between the decimals and the instances of the "books of sand" produces unique numbering for each one of the "books of sand".
Finally, what about the decimals ending in zero or the repeating decimals? What
Book of Sand will we assign to the decimal 0.20000.... or to the decimal 0.33333...? Simple, you just don’t take it into account because the
Continuum without the decimals ending in zero (the rational decimals) and the
Continuum without the repeating decimals (fractional decimals) is still a
Continuum. So, if we chose to take as domain the real numbers omitting the rational numbers and omitting the repeating decimals, we still have a
Continuum as the domain.
The mappings
F and
G are not mutually reciprocal, that is, the mapping
G that assigns a "book of sand" to a given decimal is not the inverse of the mapping
F that assigns a decimal to a given "book of sand". If that were the case they would be called one-to-one (1-1) mappings. However, the mappings need not be reciprocals, what is needed is already satisfied: for each element of the domain of each of the functions there corresponds unequivocal images on both mappings.
We have shown the correlation that exists between the
Book of Sand and the
Continuum of decimals, but the
Continuum is more than simply another term for the infinite. The
Continuum is the
Transfinite, and by
Transfinite we understand the infinite that it is not correlated with the infinite we normally use and understand.
That is, the "thought experiment" that we started at the beginning of the second article consisting in enumerating all the possible combinations of pages of the
Book of Sand, is not possible to be carried out NEVER for four reasons:
1. We do not have the TIME for all combinations.
2. Even if we have the time, the combinations are just more than infinite.
3. The combinations are TRANSFINITE so that NEVER can be completed even having an infinite time.
4. The nature of the
Continuum and at the nature of the
Book of Sand is such that not even passing an infinite number of pages per second and even having an infinite number of seconds we will succeed in all possible combinations of the
Book of Sand.
A surprise for Borges and Cantor
Borges knew that the book he had in his hands was infinite; it was “an impossible book”, a “monstrous book” as he describes it. A book with no beginning page and no ending page what else could it be? He thought of burning the books for the many nightmares the book had caused him, but “I feared that the burning of an infinite book might likewise prove infinite and suffocate the planet with smoke”.
But we have seen that the book he had in his hands was much more than being an infinite book: it was a
Transfinite book. Trying to destroy it will have consequences, not only for the planet but for the whole universe in every possible time and dimension.
On the other hand, Cantor discovered that the decimal numbers in the real line are much more than the simple infinitude of the natural numbers. He discovered that there are infinities that cannot be paired with other infinities in any way.
Cantor created the
Continuum and the
Transfinite and Borges found an application for them. Maybe, in an unknown space and unfamiliar dimension, they might be talking about the coincidence of the
Continuum and the
Book of Sand as we are doing it here.
Despite this book being infinite, as Borges admits (and transfinite —according to my interpretation) he also suggests that it can be equally finite. Is this possible?
This will be seen in the next post.
Archimedes is the first documented one to tackle down the needed mathematics to show that it is impossible the for the sand to be infinite. Strictly speaking, what he showed was that we can count how many grains can a universe hold, no matter how big it is.
