4DLab - Critical Thinking  

Three unexpected behaviors of the infinite

The infinity is a fascinating idea. Philosophers, mathematicians, theologians, artists, writers, children, almost everybody has something to say about the infinity. It seems that the infinity is not so alien, not so strange,  to the human mind. Somehow, it so easy to talk about the singular as to talk about the plural; so easy to talk about the one as to talk about the many; so easy to talk about the limited as to talk about the unbounded.

No matter the loose or imprecise usage we give to the notion of infinity, we all have an intuitive idea that the infinity is the unending and the limitless. Is it that the idea of infinity is inbred in our minds? How come that we constantly try to understand something that is physically unattainable? Or is it?

The infinity is often treated from three distinct aspects: from the philosophical way, from the mystical, and from the mathematical point of view. Here we will only deal with the infinity from the mathematical point of view.

This article is about three different points of view as attempts for explanations for the mathematical infinite. The first is the intuitive traditional unending infinite. The second point of view is the sophisticated infinite as a mathematical mapping of itself. Finally, as the third attempt in our venture in understanding the infinite is the literary infinite viewed as a dynamic entity that escapes and defies any explanation, even the mathematical one.

1. Infinity as an unending counting process

Aristotle stated a difference between the actual infinite and the potential infinite. For him, nothing in the physical universe can be infinite. However, the totality of the natural numbers is a potentially infinite. In the mathematical realm, he is right in the sense that no matter how familiar are the numbers for us, we will never grasp the totality of them.

It is credited to Euclid to assert later that the amount of prime numbers is greater than any pre-assigned magnitude. That is the same as to assert that the primes are infinite. Ancient Greek philosophers were busy searching for a satisfactory definition for the infinite. For Euclid, the prime numbers constituted a potential infinite.

The proof that the prime numbers are infinite

Proving that the prime numbers are infinite is simple. If they finite, then we could form a new number, say P, to be the product of all of them. Since every natural number is prime or the product of primes (composite) in a unique way, then P+1 must be prime because it cannot be factored with the factors of P. Hence, the primes are infinite.

But for the mathematically untrained people, the statement that nothing in our universe is infinite is hard to digest. If you grow in a place that is surrounded by beaches or sand, it is easy to state that the grains of sand you see up to the horizon is infinite.

Let's inquire a little into the amount of grains of sand in our planet. I've seen that for most of the people is easier to accept that the grains of sand are infinite than to accept that they are not. Somehow, we are natural believers of the infinity. In some manner we don't understand, it is easier to accept that something is infinite than to search for a proof that it is finite.

Is is it that we --at some evolutionary stage of our minds-- think of the infinity as a number, let's say, the biggest number? If the answer to this question is yes, then, we, the human beings, are not truly grasping the concept of infinite as something that is beyond every possible bound, but instead, we are are thinking of the infinite as something unattainable within our life span. In the case of the infinite-grains-of-sand-believers they must be thinking that the grains of sand in all the beaches and in all the deserts, is infinite because nobody can count all of them.

The assertion is true: it is impossible to count all the grains of sand of the planet. Who can do it? How can be done? How can we be sure that we count all of them without leaving a single one? Where are we going to put the counted grains so that we do not count not a single one twice? What about the sand that is below the sea?

The proof that the grains of sand are finite

It is physically impossible to count on a one-by-one basis the grains of sand in our planet, but it is possible (and easy) to calculate an upper bound for this quantity. This is exactly what Archimedes of Syracuse did millennia ago. In his opus The Sand Reckoner , and using his own notation system for very large numbers and quantities, Archimedes estimated that the universe --not the comparatively insignificant Earth-- could be filled with about 10⁶³ grains of sand. Of course, Archimedes used as the size of the universe, the universe known at the time.

At the time of Archimedes and Aristarchus of Samos (the former advocate of the heliocentric planetary model) the known planets were Mercury, Venus, Mars, Jupiter, and Saturn. So, basically, Archimedes estimated the grains of sand that would fill a sphere the size of Saturn! Obviously, the Earth is a much smaller sphere than a sphere with diameter the orbit of Saturn, so all the grains of sand in all seas and ocean beaches and deserts in our planet is a quantity significantly smaller than the amount of grains that can hold a sphere the size of the universe (his universe).

10⁶³ is a very large number, but nonetheless finite. To write it down it takes the number 10 followed by 63 zeros. But the number 10⁶³⁰ is even larger; (it is the number 10 followed by 630 zeros!). If we keep following this line of reasoning, we will be finally obliged to admit that the grains of sand in our Earth are physically impossible to count, but theoretically numerable, hence finite in quantity. The lesson learned here is that we cannot use physical entities as ground for reasoning about the concept of the infinitude. 

The simplest mathematical explanation for the concept of the infinite is the sequence of the natural numbers. Let S₁, S₂, S₃, be sequences of natural numbers defined as follow:

S₁: 1

S₂: 1, 2

S₃: 1, 2, 3

S₄: 1, 2, 3, 4

S₅: 1, 2, 3, 4, 5

See that if we keep adding one to the last term of a sequence, the next sequence is larger than the previous, so there is no way of finding the last natural  number. In summary, we can definite the mathematical infinity as follows:

S_∞: 1, 2, 3, 4, 5, ...

Contrasting the results of Euclid's infinitude of primes with Archimedes' finitude of grains of sand, is is easy to see that there are more prime numbers than grains of sand in the universe.

We now arrive at the first of our three definitions of the infinite: infinity is the unending sequence of the natural numbers¹.

2. Infinity as a mapping of itself

Infinity, taken as an unending counting process is maybe the most common, and most intuitive approach to this idea. But there is also another point of view to approach the infinity concept.

The second approach to infinity is based on the concepts of sets, and (1-1) mappings between two set of numbers. Sets are collections of objects, and a mapping² is a unique assignment from the elements of one set to elements of another set.  For our purposes, set are collections of numbers.

The concept of sets as collections of elements and their properties and implications for the study of the infinitude was deeply investigated by the German mathematician George Cantor (1845-1918). The theory of set is a beautiful field of mathematics, but quite complicated when dealing with infinite sets. However, there are some simple but astounding results that are useful for our purposes.

Sets and proper subsets

Suppose that, as above, we denote by S_∞ the set of all natural numbers, and by S_e the set of all even numbers.

S_e = {2, 4, 6, 8, 10, ...}

At first glance, we notice that the set of all even numbers is a proper subset of the set all natural numbers. By S_e being a proper subset of S_∞ we mean that every element of S_e is contained in the set S_∞, but some elements of S_∞ are not contained in S_e. For example, no odd number is at the same time an even number, so no odd number can belong to the set of the even numbers.

Hence, we are partitioning the set all natural numbers in two proper sets: the even, and the odd. See that we are splitting an infinite set in two infinite sets. But no matter this split, S_∞ and S_e  can be matched (1-1) perfectly as follows: 1 is matched with 2, 2 is matched with 4, 3 is matched with 6, 4 is is matched with 8, etc. Therefore, to every element of the natural numbers set corresponds a unique even number, and to every even number corresponds a unique natural number.

So far everything is nice and easy; anybody can say that we are doing nothing beyond counting the even numbers, but there is more than the eyes see, because we are (1-1) pairing a set with one of its proper subsets. This cannot be done with finite sets.

For finite sets there is what is called the power set of a set. The power set is the set of all possible proper subsets of a collection. To illustrate it with a simple example, let A be the following set of 4 elements:

A = {3, 7, 24, 2.5}

Then there are 15 possible proper subsets of A. Among them are: {7, 2.5}, {3, 24, 2.5}, {3}, etc. But there is no proper subset of A with 4 elements (repeating an element is not considered). So we will never be capable a (1-1) matching A with any of its proper subsets.

Curiously, infinite sets exhibit the following behavior: some infinite sets can be matched with at least one of their proper subsets, but not every infinite set can be (1-1) paired with at least one of its proper subsets.

The case that best exhibits this odd behavior is the set of all real numbers.  In simple terms, the real numbers are those common decimal numbers, like 0.9999..., 3.14151..., 0.33333..., etc. In every day life, the preceding examples are just the numbers 1, π, 1/3, etc. That is, the real numbers are all the natural numbers, all the irrational, all the fractions, etc. Therefore, the natural numbers are a proper subset of the real numbers.

The real numbers and the natural numbers are clearly infinite sets, however, it is impossible to establish a (1-1) correspondence between the reals and the naturals, but this doesn't make them finite. It is a conduct similar as the one exhibited by the finite sets. Not because infinite sets are impossible to (1-1) match among themselves, but because the set of the real numbers is a transfinite set. That is, the naturals and the reals are distinct types of infinites.

This hierarchy of infinitudes was put forth by George Cantor. He found that contrary to the finite sets where we can specify only one category of finiteness, for the infinite sets there is an unending ladder of infinitudes. For this reason, we use different symbols for the infinity:

∞ is loosely used for the infinity concept,

₀ denotes the first type of infinite, i.e., the infinitude of the natural numbers,

₁ denotes the first type of transfinites, i.e., the infinitude of the real numbers.

The second definition of infinite it that only with infinite sets we can (1-1) match the whole with one of its part.

3. Infinity as an indefinable process.

In a strange short story titled The Book of Sand, Jorge Luis Borges, the famous Argentina writer exposed a unique view about the infinity. In the story he exchanges an antique Bible from his collection (and some money) for a book that a stranger offered him. The book he received had three odd behaviors when opened no matter how many times he opened the book:

No figure in the book appeared twice, never.

"It also bore a small illustration, like the kind used in dictionaries--an anchor..."
"It was at this point that the stranger said: 'Look at the illustration closely. You'll never see it again"

The book reshuffles its page each time is closed. When reopened, it is a new book ... a new infinite book

"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 small illustrations, I verified, came two thousand pages pages apart. .... Never was an illustration repeated."

The first and last page of the book were always different.

"The stranger asked me to to find the first page. I laid my left hand on the cover and, trying to put my hand on the flyleaf, I opened the book. It was useless, Every time I tried, a number of pages came between the cover and my thumb. It was as if they kept growing from the book. Now find the last page. Again I failed. In a voice that was not mine I barely managed to stammer. "This cannot be". Still speaking in a low voice, the stranger said, It can't be, but it is.

It can't be, says Borges, but it is (he emphasized the word). Is Borges going against Aristotle when the last one said the actual infinite is impossible? Borges is saying --in better literary words-- yes, it is possible to hold an infinite object in our hands, if we hold some parts of it at a time.

The numbering of the pages was always random.

"I noticed that one left-hand page bore the number (let us say) 40,514 and the facing right-hand page 999. I turned the leaf; it was numbered with eight digits."

The number of pages in this book is no more or less than infinite. None is the first, none the last.

The numbering of the pages of The Book of Sand were random infinite series like this ones:

{..., 76, 5611190, 40,414, 999, 4577192003, ...}

{..., 71884399234, 56, 1, 2914, 2077013, ...}

Since all the pages were numbered randomly, then at some instances the numbering of the pages were all ending in 5, like this

{..., 1325, 92735, 98745, ...}

and at some other instances the pages could numbered by prime numbers only:

{..., 1601, 3, 10463, 2, 73013, 7, 5, ...}

In every case, all the numberings were merely infinite series of natural numbers without beginning, without end, without ordering. This is a contrast with our intuitive perception of the infinite since we assume the infinite to have no ending, but somewhere it must start. The best example is the series of the natural numbers we have so much used.

However, the set of the real numbers exhibits a similar behavior: for us it is impossible to determine where the positive real numbers start. This is equivalent to ask the following: which number is the positive real nearest to zero? No way to find it  Which number is the greatest positive real? No way either.

Simpler than that, the segment of the real numbers line from (0 to 1) is infinite in content. If we leave out the 0 and leave out the number 1, we still have an infinite set of points without a definite beginning and definite end.

The third definition of infinite: every infinite is an instance of itself.

E. Pérez
jul-10

References in print:

Borges, Jorge L. (1978). The Book of Sand. E. P. Dutton. New York.

Notes:

For a historical account of the philosophical basis see Infinity in the The MacTutor History of Mathematics archive.

See also in this Website the article: Where does infinite begin?

Footnotes:

[1] We are leaving out the number 0 as part of the natural numbers set without any loss of generality.

^[2] A deeper understanding of sets, functions, and (1-1) and mappings can be found in my free EBook Foundations of Transcomplex Numbers that can be downloaded from this Website.