The numbers are infiniteit is something that we all know from an early age since we understand that when we count we can go on without ever stopping. But … how infinite infinite is? The infinite, expressed by the symbol ∞, is a concept and a mathematical abstraction that indicates an unlimitedly large size. You must know they exist larger infinite And smaller infinite. On two feet, it would be wondering how it is possible that something that never ends can be smaller than something else.
We try to understand it starting from the first mathematical thing we learned to do: count! It is in fact trying to count, or rather number, The infinite sets that we discover its properties and also that there are infinitely large sets that refuse to get counted.
Counting means ordering and numbering
Let’s imagine we have to count the two piles of balls of the figure to establish which of the two contains more balls.
So at first glance it is not immediate to be able to do it, but if we move the balls of each pile e We put them in lineone after the other, we can then number them in a growing way following the order of the row, as in the figure below. Once finished, the latest number of each row tells us exactly how many balls are in the pile and we can thus compare them easily.
Using a slightly more mathematical language we can say that we have numbered The two piles using numbers 1, 2, 3, 4 … or the numbers that are used to count and that mathematics call numbers Natural. However, it was not so simple and to be able to do it we first had to put the balls well in order one after the other. With the balls it is all easy but if we tried to count the grains of sand of a beach, these would escape us continuously by hand making us lose their account. Here in mathematics with infinite sets something similar happens: there are some that can be counted, like the balls, and others more capricious than, like sand, do not get counted!
What does it mean to count an infinite set?
But what does it mean to count an infinite set? We see it with the simple case of equal numbers. For “count them” Enough put them in order – for example following the scoreboard of the 2 as we did in the figure below – and then mark each of them with a natural number Starting from 1
If we imagine we can go on indefinitely, each equal number will be marked with a natural number. For example, the equal number 1000 will be the 500 ° number that we count and so on, there will not be one will escape even one: for this reason it is said that the equal numbers are numerable. In Matematica if a whole is called Numerable, We can count the elements, that is, we are able to Put them all in order marking them one by one with a natural number.
The larger infinites cannot be counted
But if i equal numbers you can count, Like the balls, which one can be so large that it cannot even be counted? In reality, you don’t have to go so far to find an example. The German mathematician Georg Cantor In the twentieth century he showed that i decimal numbers between 0 and 1 They are so big that they cannot be counted. Meaning what
The set of decimal numbers between 0 and 1 is an infinite larger than natural numbers.
It seems incredible, right? To understand this concept, let’s try to count decimal numbers between 0 and 1.
The trick, as we have seen, is to be able to put them in order, that is, lined up one after the other. Any order is fine, it doesn’t matter that they are in order of magnitude, we are not measuring them, but only counting! It is not easy, but we imagine that we have succeeded, we will then be able to write a Infinite numbered list that contains them allbut everyone. We would therefore have a 1st number of the list, a 2nd a 3rd and so on, something like:
- 0.3713802846156317 …
- 0.7612323985243269 …
- 0.9348795729723579 …
- 0.4987526363532986 …
- ….
With a little imagination we can imagine that our list contains all the decimal numbers between 0 and 1, none excluded. So far easy, but now a procedure invented by Cantor comes into play, thanks to which we can start Build a new number of the same type, which however is not found in the list. Let’s proceed like this:
- As 1st figure after the comma we take a random figure, for example, 9 that it is different from the 1st figure of the 1st number (3) So our number will certainly be different from the 1st number of the list;
- As 2nd figure after the comma we take 7 which is different from the 2nd figure of the 1st number (6) So our number will certainly be different also from the 2nd number of the list;
- As 3rd figure after the comma we take 1 which is different from the 3rd figure of the 3rd number (4) So our number will certainly be different also from the 3rd number of the list;
- We continue this procedure indefinitely.
Our new number begins with 0.971…, has an infinite number of figures, and is different from all the list numbersso even if we could control the whole list we would not find it: in short, it is a grain of sand that has escaped us from the hands! Of numbers like this, we can build infinite, just repeat the procedure by choosing, step by step, always different figures.
We started with a list that contained everyone (but everyone!) The numbers with infinite figures after the comma, and we found ourselves having in hand an infinite number of new numbers that are not found in the list, the same list that should have contained them all: in practice these numbers are so many that it is impossible to put them in line one after the other and therefore It is impossible to count themwe couldn’t do it even if we had eternity available! Here we have discovered a capricious amount that refuses to get counted and that in mathematician is said Not numbers: we discovered a new infinite, bigger than that of natural numbers!
