Paradoxes of the Infinite

The "Gabriel's Horn" is an ideal object
that can be finite and infinite at the same time.

Can an "object" be finite and infinite at the same time? Contrary to what our intuition dictates, it seems that this duality can arise in mathematics.

We are used to thinking that the "infinite" is a well-defined concept like some of our everyday ideas of "here", "there", etc. But, if that were the case, we would not have so many paradoxes arising from this field of science that Gauss referred to as the "queen of sciences".

Of course, George Cantor did a great contribution fixing the traditional and loose understanding of the infinite, especially introducing a scale of infinitudes when he proved that there is more than one kind of infinite. Since then, many other mathematicians and philosophers had been kept busy untangling some paradoxes that the new scale of the infinite had brought.

Among them, Jorge Luis Borges surfaced and worked literally with great success the problem of random infinite series without a first and the last term.

To learn about Borges' unusual understanding of the infinite follow this series of posts beginning with the post Nobody understand the infinite so well as Borges.

To learn about the unusual paradox of how can an "object" can be finite and infinite at the same time follow this article Top myths about the infinite about Torricelli's trumpet also called Gabriel's horn.

Is the Book of Sand a Book from the Fourth Dimension?

The Book of Sand is a short story
by Jorge Luis Borges.

This is the fourth article in the series about the popular short story The Book of Sand of Jorge Luis Borges.

I finished the previous article with the following paragraph:

"Despite of 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?"

The goal of the current article is to show that the book Borges bought from a Bible salesman was a mysterious object that was more than infinite in content: it was also a multidimensional and extra-real object, a supernatural book that was capable of existing in several dimensions, and being finite and infinite at the same time.

The Book of Sand is a hyperbook

What are some of the hints that Borges give us that point toward the book extending to other dimensions beyond our 3-dimensioned space? There are two:

• At the beginning of the story, Borges mentions his initial intention of starting his story as a geometrical plot. We see this in the very first paragraph where he writes: "The line is made up of an infinite number of points, ... , the hypervolume of an infinite number of volumes". And then proceeds "No, unquestionably this is not—more geometrico—the best way of beginning my story". With this brief introduction he follows the geometrical tradition of Charles Hinton (1853 - 1907) with his books and essays about the fourth dimension like What is the Fourth Dimension, and Scientific Romances, and Edwin Abbott Abbott (1838 - 1926) with his Flatland: a Romance of Many Dimensions —a journey to the 1 and 2-dimensional space and a journey to the fourth dimension. Borges, like them, constructs the dimensioned space as a series or set of infinite points, planes, and volumes, but nowhere else in the story, he makes a reference again to the concept of dimensionality. However, and in analogy with those concepts, we will shortly see that he implicitly assumes that a hyperbook can be made up of infinitely many books.

But, what is a hyperbook? Let us first review what is hypercube. Similarly to the line, the plane, and the cube, a hypercube is a cube that is the result of moving or dragging our ordinary cube to the 4th-dimension. But where is the fourth dimension? Although difficult to imagine, the fourth dimension is supposed to be "perpendicular" to our ordinary three dimensions. We (as Borges) arrive at this conclusion by simple induction: the plane is generated by moving a straight line perpendicularly to its length; a solid is obtained by moving a plane perpendicular to its surface. Note that in each case the motion is perpendicular to the dimensions of the previous object.

The hypercube is similarly obtained: by moving a cube perpendicular to the three dimensions it has. Visualizing the hypercube is not an easy task because we cannot imagine where is a dimension that is perpendicular to our daily 3-dimensional world. Note that according to the previous assumptions a plane is a "hyperline", and a cube is a "hyperplane". Furthermore, even a cube can be considered as a "hyper-hyperline", but since this concept is little more than meaningless, we can plainly say that the cube is also a "hyperline".

Returning to our hyperbook, let us simplify the shape of the book assuming that it is like a cube (a book can be cubical). Then, a hyperbook is a 4-dimensioned book such that any reduction of its 4-hyperspace to a 3-dimensioned space results in an ordinary book.

Mathematicians not only speak of hyperspaces and hypercubes but also of hyperspheres, so Borges —who was also related with modern mathematics— simply extended this concept to the books: if mathematicians could conceptualize such hyper-objects, for him was also very easy to conceptualize the hyperbooks.

Attributing to Borges the idea that The Book of Sand is a hyperbook —and not a simple solid— is not a sound reasoning if we cling to the above arguments alone. But there is another passage in his story that reinforces my argument that for him The Book of Sand is not a "mere" infinite book: it is also a book from other dimensions.

Let's quote again the passages to where I'll make reference:

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, as if by a schoolboy's clumsy hand.

It was at this point that the stranger said. "Look at the illustration closely. You will never see it again."

Borges noted the place and closed the book, but once he reopened it he never found again the illustration of the anchor.

However, later he found another illustration: a mask. But there was a curiosity among them and he explicitly narrates it for us:

The small illustrations, I verified, came two thousand pages apart.

A book with illustrations every 2000 pages? Take notice that he is not saying that the next illustration is so far: he is saying that all illustrations are so far apart. He is not explicitly referring to those two illustrations, the anchor and the mask; so we can safely assume that he is writing about all the figures of the book.
Isn't this crazy? He says he verified this fact of the illustrations separation, but what type of book of has this particularity? Why exactly 2000 pages apart?

Borges used an alphabetical notebook to record the pictures he found:

I set about listing them alphabetically in a notebook, which I was not long in filling up.

The book was somehow full of illustrations because —as he says — it didn't take him too much time to fill the notebook, despite "Never once was an illustration repeated".

The Book of Sand is a dictionary

Remember the quote: "... a small illustration, like the kind used in dictionaries"? From this quote we obtain the second hint: The Book of Sand is a dictionary! This marvelous book —this hyperbook— is a dictionary because:

• It never repeats an illustration: every instance of a "book of sand" is just a definition of an object! This is what dictionaries do: they show a single small plate or diagram and then a short or long explanation of what is this object. Every time the book is reopened, a new random definition with its corresponding pictogram appears.
• Only a single picture per book "instance" appears. Every time he opened the book the whole book is dedicated to the description of the picture he found. Sadly, he didn't understand the accompanying definition and prose because "The script was strange to me".

The dictionary was unreadable for Borges because it was written in a foreign language or dialect: "It seems to be a version of the Scriptures in some Indian language, is it not?" he asked the salesman, but the answer was "No ... I acquired the book in a town out on the plain in exchange for a handful of rupees and a Bible. Its owner did not know how to read. I suspect that he saw the Book of Books as a talisman".

The Book of Sand is a dictionary: what a surprise! No wonder the figures were 2000 pages apart; each picture definition was 2000 pages long so that each "copy" or "instance" of the Book of Sand is dedicated to one particular object or thing. Now we understand why the figures were never repeated: because each "book of sand" consisted of a 2000-page long definition.

The Book of Sand can be interpreted
as book from higher dimensions.

The Book of Sand as an object of parallel universes

Is the possibility of multiple instances of the same object in multiple dimensions coexisting together an insane idea? Not at all, because with the triumph of the modern physics the concept of multiverses is just one of the many hard ideas to digest.

But, how and why so many "instances" could occupy the same time-space in such a manner that the book could be held in Borges' hands? The answer is simple: remember that a hypercube is a cube surrounded by cubes in every possible dimension; therefore, in a similar manner, The Book of Sand is a hyper-dictionary surrounded by dictionaries in every possible dimension. In this way, every time Borges opened the book he was opening a dictionary in other dimensions. This is the reason why he could hold a multidimensional infinite dictionary in his hands: he was holding only a 3-dimensional instance of and infinite-dimensional dictionary. All other "copies" or chapters, or "definitions" or "instances" were in very near dimensions: just touching the "real" one, but inaccessible at the same time, as the figure at right shows.

The notion of parallel universes is not more insane than the notion of a single an unique universe; both extremes are hard to understand. I leave the reader with two simple questions related with the parallel universes idea: What law of physics states that there should be a unique 3-dimensioned space? What law of physics states that there cannot exist more than three dimensions?

The 10 Top Myths About the Infinite

Myth 1: An infinite split by one half is no longer infinite

Let's us take the set of all natural numbers, i.e., the numbers we use to count, like 1, 2, 3, ... We will represent this set by the symbol Z. Each one of the natural numbers is either odd or even; the odd numbers being 1, 3, 5, ... and the even numbers 2, 4, 6, ... Note that the numbers we call even are those divisible by 2. Hence every natural number is either divisible by two or not. Those that are not divisible by 2 are the odd numbers.

Natural numbers = odd numbers + even number

Z = {1, 2, 3, 4, 5, 6, 7, ...} = {1, 3, 5, ...} + {2, 4, 6, ...}

The even numbers are infinite because there is no end to this series. Same with the set of odd numbers: there is no way to find and end to this series. So the infinite set of all natural numbers is the sum of two infinite series; the series of the odd numbers plus the set of the even numbers.

If you take away the infinite set of the even numbers from the infinite set of the natural numbers you are left with an infinitude of odd numbers.

{1, 3, 5, ...} = {1, 2, 3, 4, 5, 6, 7, ...} - {2, 4, 6, ...}

To a similar behavior we are faced if we take away the set of the odd numbers from the set Z.

Therefore, it is not necessarily true that if we split an infinitude in a half, the two parts are no longer infinite.

Myth 2: One infinite added to another infinite is a greater infinite

This one is the opposite of the above myth.

Myth 3: If we increasingly take away infinite elements from an infinite set, eventually, the remaining set is no longer infinite

This is not the same as Myth 1: there we were linearly taking away one integer for each one left.

Suppose that to the set of all natural numbers Z we remove numbers from it using this pattern:

1. Leave the number 1, but take away the next 2. We are left with {1, 4, 5, 6, ...}
2. Leave the number 4, but take away the next 5. We are left with {1, 4, 10, 11, ...}
3. Leave the number 10, but take away the next 11. We are left with {1, 4, 10, 22, ...}
4. Repeat the pattern over and over again.

Note that with each step we are taking more an more elements away from the original set of the natural numbers. The separation between the remaining integers is wider and wider. If we repeat this process indefinitely, we'll be progressively removing more an more elements. This is far from the first example above where we were removing even or odd numbers only, because in this schema we are removing from both types of numbers.

However, no matter how far we go or how many integers we remove, the remaining set will be always infinite because although the steps are infinite, the elements to be removed are always finite.

Myth 4: There are more fractions than natural numbers

This assertion might appear to be against our intuition because we assume that since every natural number can be expressed as a fraction, like

1 = 1/1,
2= 2/1 = 2/2,
3 = 3/1 = 6/2 = 9/3 ...

we can conclude that there are more ways of expressing fractions than the numbers themselves. However, note that in the pyramidal scheme above, we can count the fractions as follows:

1/1 = is the first

2/1 = is the second, 2/2 is the third

3/1 = is the fourth, 6/2 is the fifth, 9/3 is the sixth,

Hence, no integral fraction can escape our counting scheme. Therefore, the integral fractions are countable which means that there are not more integral fractions than natural numbers.

Myth 5: An infinitude of elements multiplied by another infinitude is always a grater infinitude

Myth 6: Since every fraction can be converted to a decimal then there are as my decimals as fractions

Myth 7: The segment of the line from 0 to 1 contains double the points as the segment from 0 to 1/2

Myth 8: The number of grains of sand is infinite.

This is a classic myth. Probably all of us, at some stage of our live, had think that the grains of sands are infinite.

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.

At his time the observable universe was up to Saturn, so what he did was to compute how many grains can fill a sphere the size of the orbit of Saturn. The mathematics needed to arrive at his conclusion were simple, but ingenuous extensions he devised for the arithmetic of his time was an enormous contribution.

You can download his all-time famous book The Sand Reckoner here.

Myth 9: If a vase is infinitely long, then it must have an infinite capacity

This is a beautiful one ...

Gabriel's Horn


Myth 10: If there were infinite universes out there, in some of them, or at least in one, should be an exact copy of our planet Earth

The Book of Sand Is a Transfinite Book

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 ≠ t_1, 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 t₁ and t₂ in the examples above? Well, since t₁ = {... 40, 51, 4999, ... } and t₂ = {... 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 S_b be any sequence of page numbers of some Book of Sand with elements e₁, e₂, e₃ ... etc. that is, S_b = { e₁, e₂, e₃ ... }. Note that this is equivalent to say that the page numbers are e₁, e₂, e₃ ...

Let N_b be the product of all the primes with positions p_e1, p_e2, p_e3, etc. That is,

N_b = p_e1 × p_e2 × p_e3 × ...

Let D_b be the decimal obtained when we add a decimal point in front of N_b. Suppose the decimal digits of D_b 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 (S_b) = 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 S_b 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 D_b 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 (S_b) must have an associated number N_b = 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 (S_b) 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 d_c be a nonrational decimal chosen between 0.1 and 1.0. Let d₁, d₂, d₃, ... be the digits of this decimal. Let n_i be natural numbers made by the digits of this decimal as follows:

n₁ = 10¹ + d₁

n₂ = 10² + d₂ × 10 + d₃

n₃ = 10³ + d₄ × 10² + d₅ × 10 + d₆

Therefore, to the decimal d_c made of the decimals d₁, d₂, d₃, ... corresponds the set made up of the above sequences: S_n ={[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(d_c) = 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.

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 with 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 infinite 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 infinities, therefore, we must invent new categories for certain sets. These sets were called by Cantor Transfinite; infinities way beyond the infinity of the natural numbers.

The simplest example of transfinite sets is 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 number 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 enumerated 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.

There are as many instances of "The Books of Sand"
as there are real numbers.

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 an 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.

What Is the Shape of a Wheel in the Fourth Dimension?

In 1909, the renowned magazine Scientific American held a contest where authors were called to submit articles answering the question "What is the fourth dimension?" The magazine received more than two hundreds essays, a respectable quantity for such an obtuse subject at that time.

The judge in charge to select the best articles was Dr. Henry Parker Manning (1859-1956), a mathematics professor at Brown University. Manning was a specialist in non-traditional geometries and algebras like non-Euclidean geometry and quaternions.

One of the rules of the contest was that the articles should not be greater than 2500 words; thus the essays were going be medium sized in length. Another rule was that the essays should be submitted with pseudonyms instead of the true author name. Since each author was writing independently of the others, and from different countries, some repetitions in concepts were inevitable.

Readings of The Fourth
Dimension Simply Explained

Out of the large amount of essays, Dr. Manning edited a book of what he considered the best 22 articles, and wrote an Introduction for them where he exposed his view of some of the articles selected, and even corrected some misconceptions about transformations and manipulations of objects in the fourth dimensions, like turning gloves inside-out. The book was published under the title: The Fourth Dimension Simply Explained.

Reproduced below is the discussion of Manning about what should be a wheel in four-dimensional space.

A wheel of four-dimensional matter, in two dimensions of the shape of a circle and in the other two dimensions very small, would have for axis a flat plate instead of a rod. This axial plate could extend indefinitely in all the directions of its plane without any interference with the wheel. The wheel can slip all around over the axial plate unless held to some position on it, just as with us a wheel may slip along on its axis unless held to some position on it. We may suppose that in a three-space we can see the axial plate and a pair of opposite radii (spokes) of the wheel, appearing to us entirely separate; in this way we can see a two-dimensional hole. Or we can see the entire wheel with a hole through it and an axial rod, cut from the axial plate by our three-space.

Manning included no figures to clarify his ideas, but we can suppose that what he did is that in the same way that a line (an axis) projected into the next dimension would produce a plane, he deduced that an axis holding two wheels, when projected into the next dimension would become a plane. It is not easy to visualize two linked rotating wheels in 4D where their common axis is a plane, but anything about the fourth dimension is not easy.

But Manning goes further and writes:

We can fasten the wheel rigidly to the axial plate so that it will turn with the wheel, the wheel turning in its plane and the plate turning on itself. We may put more than one wheel on an axial plate, putting different wheels at different points on the plate wherever we please. If these wheels are all fastened rigidly to the axial plate we turn them all by turning one. Thus we have a method of constructing machinery in space of four dimensions.

If this is not enough to dazzle your mind, wait until you read this:

The axial plate may itself be a wheel. We may fasten two wheels together at their centers making them absolutely perpendicular to each other. Such a figure can revolve in two ways, the plane of one wheel being the axis plane of the rotation and the plane of the other wheel the rotation plane.

Dr. Manning should be speaking from a strictly mathematical point of view; he cannot be fantasizing about higher dimensions. However in the past article The strange extraterrestrial worlds of Camille Flammarion, in the paragraphs about the controvertible Flammarion's woodcut, I called to the attention to the enigmatic solid wheel that appears at the top of the "woodcut" (the woodcut figure is repeated here). Note how in this woodcut, the two intersecting wheels are drawn like two classic ox cart wheels. Possibly, when the artist carved this --let's call it, cross-wheel, or super-wheel-- he was not thinking about a fourth dimension, he needed not to. What he tried to convey was the idea that beyond the spheres that limit our imagination many things can coexist even when they appeared to be contradictory to our senses. Hence, for this artist, wheels that can move in two directions simultaneously are possible. Manning, speaking without the need to recur to metaphors tells us that this is possible; in a 4-dimensional world.

Adding to his exposition of a 4D-wheel, Manning says:

We might have a spherical wheel; something in three dimensions of the shape of a sphere and its fourth dimension very small. Such a wheel with a one-dimensional hole through it may turn on an axial rod, but its motion is not confined to a definite direction of rotation as is the case with the flat wheel turning in its plane.

 Ezekiel's enigmatic impossible wheels.

Flammarion's woodcut is not the only picture that incorporates a possible 4D-wheel. See that in the next picture there is also the same 4D-wheel element incorporated as part of Ezekiel's vision. In fact, the origin of this idea or metaphor comes from the following verses (Chapter 1 of Ezekiel 15-18 ) of the book of Ezekiel in the Bible:

"As I looked at the living creatures, I saw a wheel on the ground beside each creature with its four faces. This was the appearance and structure of the wheels: They sparkled like topaz, and all four looked alike. Each appeared to be made like a wheel intersecting a wheel. As they moved, they would go in any one of the four directions the creatures faced; the wheels did not change direction as the creatures went. Their rims were high and awesome, and all four rims were full of eyes all around."

Continuing with Manning's Introduction see the following"

A spherical wheel may be used for vehicles. If four dimensional beings lived on a four-dimensional earth; that is, alongside of its three-dimensional boundary, a vehicle with four or more wheels of either kind could be used in traveling over this earth. With a flat wheel he could travel only in a straight line without friction between the wheel and the earth; with a spherical wheel he could travel in any direction in a plane without such friction, but would meet with a slight friction in turning from one plane to another.

Download the free ebook: Readings of The Fourth Dimension Simply Explained.

We have discussed so far the Introduction that Dr. Henry P. Manning wrote to the book The Fourth Dimension Simply Explained. The edition that Datum is giving for free contains many essays about the fourth dimension that you will surely enjoy. Download it now!

Selected Puzzles from Henry E. Dudeney

Henry E. Dudeney (1857-1930) was an English logician and mathematician that specialized in creating and collecting puzzles.

Two men arguing about how much liquid
is in a closed barrel without opening it.

Amusements in Mathematics, published by 1917 is a great collection of geometrical, chessboard and magic square problems.

The free E-book that I am offering you now is titled: 44 Selected Puzzles and Pastimes from Henry E. Dudeney. The puzzles are a hand-picked selection of the most "attractive" puzzles in the sense that some of Dudeney's problems are verbal, others are about chessboards, dominoes, etc., but this selection is all about those with some graphical appeal.

Download the free ebook: 44 Selected Puzzles and Pastimes from Henry E. Dudeney for free.
Click here

As an example of the EBook content take The Barrel Puzzle, one of the Dudeney's ingenious puzzle included in the EBook.

The puzzle goes like this

The men in the illustration are disputing over the liquid contents of a barrel. What the particular liquid is it is impossible to say, for we are unable to look into the barrel; so we will call it water. One man says that the barrel is more than half full, while the other insists that it is not half full. What is their easiest way of settling the point? It is not necessary to use stick, string, or implement of any kind for measuring. I give this merely as one of the simplest possible examples of the value of ordinary sagacity in the solving of puzzles. What are apparently very difficult problems may frequently be solved in a similarly easy manner if we only use a little common sense.

Clearly, this is a situation we may encounter some day. He clearly says that the content of the barrel does not matter: it can be oil, petroleum, wine, or—as he says—water. Therefore, the problem is related with the geometry or configuration of the barrel. If the barrel need not be opened, then by the common sense he mentions, the only possible action is to tilt the barrel to "see" what happens.

There lies the solution: we have to slowly tilt the barrel until the liquid level is in "correct" position. In his own words:

All that is necessary is to tilt the barrel as in Fig. 1, and if the edge of the surface of the water exactly touches the lip a at the same time that it touches the edge of the bottom b, it will be just half full. To be more exact, if the bottom is an inch or so from the ground, then we can allow for that, and the thickness of the bottom, at the top. If when the surface of the water reached the lip a it had risen to the point c in Fig. 2, then it would be more than half full. If, as in Fig. 3, some portion of the bottom were visible and the level of the water fell to the point d, then it would be less than half full.

This method applies to all symmetrically constructed vessels.

44 Selected Puzzles and Pastimes.

Download this EBook; all the problems are illustrated and perhaps you can make your own modifications.

Hello World!

Hello, world is a venerable phrase in computer programming. It is a tradition that when learning a new computer language, the first program is the one that outputs the phrase "Hello world".

If you are lucky and brave enough to make that odyssey, you are near to become an "extremely skilled and accomplished" programmer. Well, since every computer language has its own syntax and grammar, once you know one language all you need is to get related with the new language grammar and syntax ... it seems so simple, but it's not!

Fortunately, this post is not a computer program, it is a welcome message; but the phrase will be recycled to welcome my readers about some topics in the history of science, philosophy, and the fourth dimension.

Stay around, and contribute with your comments or EMail: surely I'll learn a lot from the outside world. That's why I also say: Hello world!