There are infinitely larger than others

To some questions about quantities, for example how many candies they want or how much they love someone, children often answer with conviction “Infinity!”. And sometimes there is someone who counters “So I say infinite plus one!”, Imagining that they are expressing the idea of ​​something even more extensive. In reality, there is no difference between the two answers, but this does not mean that there are no infinities greater than others. It is a concept that mathematicians have (literally) dealt with for a long time and that even philosophers have puzzled over, demonstrating how difficult it is for our finite minds to deal with something we can barely conceive.

To try to get at least an idea, it is better to start with something small and circumscribed and with sets. As the word suggests, a finite set is a collection of objects, or to say elements, that can be counted: an enclosure with 32 sheep, for example.

Determining the size of a finite set is quite simple, as you just need to count the elements it contains. We also know that sooner or later we will finish counting, because the whole is finished. Things get complicated when we are dealing with an infinite set, where we could spend our whole life counting without finishing before taking our last breath (the set of our breaths is inevitably finite).

The classic example used in these cases is that of natural numbers (ℕ), that is, the numbers we use to count and order:

ℕ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 …}

Since there is not the largest possible natural number, trying to count all the elements in the set ℕ would get us nowhere. This is why we say that this set is infinite, but defining it as such can be a little misleading. Basing our idea of ​​infinity exclusively on this characteristic of natural numbers makes it more difficult to imagine that other infinities exist, and that they can be conceptually larger.

Real and rational numbers can be a good example. As we learn in school, real numbers are numbers that can be attributed a finite decimal development such as 9 and -4.3, or infinite such as the value of π or √15. Rational numbers, on the other hand, can be written as fractions (example: 2/3), with a finite decimal development (example: 4.3) or with an infinite but periodic one (example: 4/3 which is 1.3333 …) . Every natural number is also a rational number, and it can be shown that despite being “more”, rationals are infinite like naturals, that is, I can count them. The same is not true for real numbers, to the point of being able to define them as a different infinity.

Between the numbers 2 and 6, for example, we can insert only a finite set of natural numbers: 3, 4 and 5. And we can also insert an infinite number of rationals (but always with periodic decimal development) and an infinite number of real numbers : √15, π, etc. And between each of these real numbers there may be still others, indefinitely.

At the end of the nineteenth century, the mathematician Georg Cantor wondered for a long time about the characteristics of these infinite sets and came to the conclusion that they had different dimensions. Many mathematicians of the time did not take it well, and Cantor had to wait a long time before his theories were seriously considered.

In fact, it was Cantor who systematized set theory as we know it today. He proved that given any set Xthere exists the set of all possible subsets of Xtogether called power of X that is P (X). And it was always Cantor who demonstrated that the whole is the power of an infinite whole X It is greater than X same. From this it follows that there is an infinite hierarchy of magnitudes of infinite sets, which makes possible the concept of cardinal and ordinal (transfinite) numbers.

When we use natural numbers to count the elements we are using cardinal numbers; if, on the other hand, we are establishing an order, we are using ordinal numbers, which therefore allow us to indicate what position a given element occupies. To say that there are 16 teams in the standings we use the cardinals, to say that Juventus is seventh in the standings we use the ordinals.

Before Cantor, mathematicians had been confronted with the Aristotelian approach according to which the infinite was definable as “potential / actual” and that of Euler as a “formal” infinite. When Cantor’s theories were finally accepted, he went on to conceive of infinity as something measurable, and to contemplate the existence of different types of infinity.

As we saw earlier, the set of real numbers has a cardinality (a magnitude) greater than the set of natural numbers. But following Cantor we can go further and show that the set of even numbers has the same cardinality as the natural numbers of which the primes are part: this means that, formally, a part (the set of even numbers) is as large as the integer (the set of natural numbers). To indicate the cardinality of a countable set such as that of natural numbers, Cantor chose a Hebrew letter: ℵ₀ (reads Alef-zero).

If you are still reading here and your head is spinning infinitely, we can try to mentally visualize a part of these concepts with a beautiful paradox, which could prove enlightening or make you sink further into the abyss of mathematics, a fascinating experience.

Let’s imagine that there is a hotel that has an infinite number of rooms and that they are all occupied. One evening a new customer comes to the reception, who has not booked and would like to spend the night in the hotel he has heard so much about. Like all hotel managers, the one in our example also struggles to find a place for the new customer and finds a solution. He turns on the communication system in the rooms and invites all guests to move to the room with the next number after the one they are in. The guest on 1 goes to 2, the guest on 2 goes to 3, the guest on 3 goes to 4 and so on ad infinitum. In this way, room 1 is free and the newcomer can sit down, and the director’s choice is the demonstration that 1 + ℵ₀ = ℵ₀.

A short time later a bus arrives at the hotel carrying an infinite and countable number of new customers. The manager who knows a lot about mathematics activates the internal communication system again and tells all guests to move to the room that has double the number of the one they are in: who is at 1 goes to 2, who is at 2 goes to 4, whoever is on 3 goes to 6 and so on. By doing so, the manager frees all the odd rooms, which are infinite and which can therefore accommodate the infinite busload of new customers. In this case the director proved that 2 · ℵ₀ = ℵ₀.

Such an efficient hotel attracts great attention and a short time later endless buses with an infinite and countable number of customers begin to arrive. The director remembers that prime numbers (natural numbers greater than 1 that have only 1 and themselves as divisors) are infinite and the new arrangements are based on this. All guests already present in the hotel must move to the room corresponding to 2 (the first of the prime numbers) raised to the power of the number of the room in which it is located. The guest in 9 must go to 2⁹i.e. to room 512.

The manager then goes to the customers of the first bus and says that each of the occupants will have to enter the hotel room with a number equal to 3 (the second prime number) raised to the power of the number of the seat on which he is sitting. The customer in seat 9 will then have to go to room 3⁹, that is to 19683. The director assigns the next prime number to the second bus – 5 – with the same instructions for raising the power with respect to the seat occupied. The third bus will have 7 as its starting point, the fourth on 11, the fifth on 13 and so on ad infinitum, since the prime numbers are infinite. Since the base is a prime number and the factor is a natural number, there can never be two assignments with the same number, thus guaranteeing a room for each new customer.

The strategies used by the director work because they are closely related to the lowest possible level of infinity. And this brings us back to the concept of cardinality of the countable: ℵ₀ by Cantor.

To accommodate existing guests and new customers, the manager used only natural numbers. If he were to deal with a larger order of infinity, such as real numbers, his strategy would no longer work because he would have no way to systematically understand each number. The number of rooms in the hotel is infinite, but it is a countable infinite: there are as many rooms as there are positive integers up to infinity. If a bus arrives with an uncountable infinite number of customers (none sitting in a numbered seat and all with a different name, so to speak), the manager would always end up with someone who had not been counted and who still needs a room. .

This paradox, which we have simplified a little, was invented by the German mathematician David Hilbert precisely to show the characteristics of the concept of infinity and to help understand the orders of infinity. It is perhaps the clearest demonstration of how difficult it is for our mind, which is confronted every day with finiteness, to conceive that there can be different infinities and above all that there are mathematical conditions that determine their characteristics. It is no coincidence that the concept of infinity has fascinated many philosophers and has led over the centuries to comparisons and disputes, infinitely more complicated than this article.