Can the Infinite Be Randomized?

Is the infinite always the same? Is the infinite tomorrow the same as is today? How many infinities are there? Only one? How about if there are infinite infinities? How about if all of them are not the same, one to another?

How about if one of those infinities is not linear, without any kind of order, but random? Why not?

We have a naive and simple idea of the infinite. In the best of the cases, we think of the infinite as the unending series 1, 2, 3, ... In the worst scenario, we think of the infinite as the quantity of the grains of sand in all the deserts and beaches on the Earth.

It was Archimedes of Syracuse, more than two millennia into the past, who proved that it was impossible that the grains of sand were infinite because he was able to give a fair estimate of the grains of sand that could be placed in a sphere the size of the orbit of Saturn.

He knew that it was impossible to count grain by grain all the deserts and beaches. So it is impossible to prove that the grains of sand are finite by enumerating them one by one.

So, his approach was to establish an upper limit to the number of grains that can hold the planet Earth. This is an indirect proof that the sand cannot be infinite.

The Sand Reckoner.

Download the free EBook The Sand Reckoner, the book where he developed his proof.

But what if we have an infinite book in our hands, how big can it be? Infinite in size? Infinite in weight?. Infinite in volume?

Have you ever heard of The Book of Sand? It’s a short story about an infinite book with no beginning and no end. But this book has a finite amount of pages; how come?

The infinite is incredible and surprising! What is going to limit the limitless?

Read the article Three unexpected behaviors of the infinite and see three unforeseen aspects of the infinite.

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.

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