Some myths about the infinite
Can the infinite be briefly defined, or does it requires an infinite definition? Let us hope that the first alternative is sufficient. In that case, we'll restrict to the infinite set of the natural numbers for most of our examples because they represent the simplest case of infinitude.
The natural numbers are the integers we use daily to count like 1, 2, 3, ... etc, The aggregate of all the natural numbers is usually represented by the symbol N although sometimes it is also symbolized by the symbol Z+ to clearly state that we are leaving out the integer zero (0). The series of the natural numbers never ends because to any given number n it is always possible to add 1 to find its successor n+1. And because this series never ends is why we call it "infinite". Sometimes, this is also denoted as 1, 2, 3, ... ∞, but we must keep in mind that the infinite cannot be properly represented by any symbol, because the symbol ∞ is not an arithmetic entity that can be manipulated under the common mathematical operations.
Another way of defining the infinitude of the natural numbers is as follows: 2 is the successor of 1, 3 is the successor of 2, 4 is the successor of 3, etc. Thus, 3 is the successor of the successor of 1, 4 is the successor of the successor of the successor of 1. Hence, N is the collection —or set, or aggregate— of all the successors of 1
Therefore, the infinitude of the natural numbers is nothing more than the collection of the possible successors of the unit element one.
We cannot prove that the natural numbers are infinite; we take them as the definition of infinitude. The aggregate of the natural numbers are the "standard yardstick" to measure other infinitudes.
Giving a symbol —like N— for the aggregate of all natural numbers does not make this set more comprehensible than not naming it at all.. Every infinitude is intrinsically incomprehensible, the reason for this nobody knows, but the role of the language may be a relevant barrier because our languages —all of them, be it spoken, written, symbolic or not— is a tool for the manipulation of finite and bounded everyday objects, and therefore, the abstract relations among them is forcibly limited.
The concept of the infinite is not exclusive of the mathematics; the humankind has found applications of it to the invisible spiritual world and sphere, to the physical and visible cosmological universe, to the unbounded large, and to the submicroscopic and subatomic infinitely small. But no matter what direction of thought we follow, we can be sure that no matter the manipulations and reasoning we get involved into, nobody does really understand or grasp what is the infinite.
The concept of the infinite has always been a source of many controversies, paradoxes, contradictions, and myths in the sciences, the art, and in the literature. Here we are going to see some examples of how our minds boggle with the concept of the unlimited, the unbounded and the unreachable.
Myth 1: If from an infinite set we take away an infinite number of elements, the remaining set is no longer infinite.
Or, can sometimes the part be as big as the whole?
Let's us take the set of all natural numbers N = {1, 2, 3, ...}. Each element of this set is either odd or even; the odd numbers being 1, 3, 5, ... and the even are 2, 4, 6, ... 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
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 reach the last odd number. So, N is the sum of two infinite series; the series of the odd numbers plus the set of the even numbers.
If we take away the infinite set of the even numbers from the infinite set of the natural numbers we are left with an infinitude of odd numbers.
{1, 3, 5, ...} = N − {2, 4, 6, ...}
To a similar behavior we are faced if we take away the set of the odd numbers from the set N.
Therefore, it is not necessarily true that if we split an infinitude in a half, the two parts are no longer infinite. Of course, sometimes this can be true; we can remove an infinite number of elements from N and the remaining set is no longer infinite.
For example, let us denote by T the set of all natural numbers greater than 10 : T = {11, 12, 13, ...}. Obviously, T is an infinite set. When we subtract T from N we are left with the finite set {1, 2, 3, ...7, 8, 9}.
Myth 2: The number of grains of sand is infinite.
Or, is infinite any quantity that we are unable to enumerate?
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 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 the 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.
Archimedes starts his book as follows:
THERE are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited.
and proceeds stating a difference between being infinite and being unable to count great quantities ...
Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the earth, including in it all the seas and the hollows of the earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken.
Now Archimedes proceeds to indirectly state that you can count 1, 2, or 3 grains of sand, then it is a matter of extending any the numerical system to be able to count to any desired number.
But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe.
Next Archimedes states that size of the "universe" is a just a matter of definition since the "universe" is not an object that can can objectively sized.
Now you are aware that 'universe' is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account as you have heard from astronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premisses lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.
Now it is easy to see that this is impossible; for, since the centre of the sphere has no magnitude, we cannot conceive it to bear any ratio whatever to the surface of the sphere. We must however take Aristarchus to mean this: since we conceive the earth to be, as it were, the centre of the universe, the ratio which the earth bears to what we describe as the 'universe' is the same as the ratio which the sphere containing the circle in which he supposes the earth to revolve bears to the sphere of the fixed stars. For he adapts the proofs of his results to a hypothesis of this kind, and in particular he appears to suppose the magnitude of the sphere in which he represents the earth as moving to be equal to what we call the 'universe.'
Archimedes continues with an arithmetical arrangement of ordersand periods resulting in that he estimates the grains of sands in about 1063. This a very big number, indeed, but finite and countable as well.
No matter the course of thought Archimedes followed, one thing is clear: the grains of sand of all our seas and all the ocean beaches plus the grains of sand of all the deserts of the earth is much less than the grains of sand needed to fill a sphere the size of the orbit our planet.
Therefore, the grains of sand are not infinite.
Go to More myths in Part 2 of this article.
Notes:
See also in this Website the article: Where does infinity begin?
See also in this Website the article: Three unexpected behaviors of the infinite . 