Economy

There is infinity greater than others

To some questions about quantities, for example how many candy they want or how much they love someone, children often answer with conviction “Infinity!”. And sometimes someone is opposed to “So I say infinite plus one!” , imagining that they express the idea of ​​something more comprehensive. In fact, there is no difference between the two answers, but that does not mean that there are no greater endings than the others. It’s a concept that mathematicians have dealt with (literally) for so long that philosophers have puzzled over it, explaining how hard it is for our limited minds to deal with something we can barely conceive.

To at least try to get an idea, it is best to start with something small, specific and groups. As the word implies, a finite set is a collection of things, or to say, items, that can be counted: a container with 32 sheep, for example.

Determining the size of a finite set is quite simple, you just need to count the elements it contains. We also know that sooner or later we will finish counting, because it’s all over. Things get complicated when we’re dealing with an infinite range, where we can spend our whole lives counting without finishing before we take our last breath (our breath range is inevitably limited).

The classic example used in these cases is the natural numbers (ℕ), i.e. the numbers we use for counting and ordering:

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 items in the set ℕ won’t get us anywhere. This is why we say that this set is infinite, but defining it as such can be a bit misleading. Building our idea of ​​infinity exclusively on this property of natural numbers makes it difficult to imagine other infinities exist, and that they could 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 to a finite decimal development such as 9 and -4.3, or an infinite such as the value of π or 15. On the other hand, rational numbers can be written as fractions (example: 2/3), with a twist Finite decimal (Example: 4.3) or with an infinite but periodic number (Example: 4/3 which is 1.3333…). Every natural number is also a rational number, and it can be shown that although it is “more”, the rationals are as infinite as the natural numbers, i.e. I can count them. The same is not true of the real numbers, to the point of being able to define them as a different infinity.

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

At the end of the nineteenth century a mathematician George Cantor He wondered for a long time about the properties of these infinite sets and came to the conclusion that they have different dimensions. Many mathematicians of the time did not take it very well and Cantor had to wait a long time before seriously considering his theories.

In fact, it was Cantor who organized set theory as we know it today. He proved it in the light of any group Xthere is a set of all possible subsets of Xtogether they are called power X This is it P (X). And it was always Cantor who showed that all is the power of all infinite X bigger X same. It follows that there is an infinite hierarchy of infinite group sizes, which makes the concept of cardinal and ordinal (transboundary) numbers possible.

When we use the natural numbers to count the elements we are using the original numbers; On the other hand, if we are creating an order, we use ordinal numbers, which thus allow us to determine the position occupied by a particular element. 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 ordinal standings.

Prior to Cantor, mathematicians encountered the Aristotelian approach according to which the infinite can be defined as “potential/actual” and Euler’s “formal” infinite. When Cantor’s theories were finally accepted, he moved on to conceptualizing infinity as something measurable, and thinking that there are different types of infinity.

As we saw earlier, the set of real numbers has a greater original number (amount) than the set of natural numbers. But following Cantor we can go further and show that the set of even numbers has the same cardinal relationship as the natural numbers of which the primes are a part: this means, formally, that the part (the set of even numbers) is as large as the integer (the set of natural numbers) . To indicate the origin of a countable set like the set of natural numbers, Cantor chose a Hebrew letter: ℵ (reads a thousand zero).

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If you are still reading here with your head spinning infinitely, we can try to mentally visualize a portion of these concepts in a nice way. paradoxwhich can be useful or make you dive further into the math abyss, which is a great experience nonetheless.

Let’s imagine that there is a hotel with an infinite number of rooms and they are all occupied. One evening, a new client comes to the front desk, who has not booked and wants to spend the night at the hotel about which he has heard so much. Like all hotel managers, the person in our example is also struggling to find a place for the new client and find a solution. It operates the in-room calling system and invites all the guests to move into the room with the number following the number they are in. The guest in 1 goes to 2, the guest in 2 goes to 3, the guest in 3 goes to 4 and so on infinitely. This way Room 1 is free and the newcomer can sit, and the choice of exit is to show that 1 + ℵ = ℵ.

A short time later, a bus arrived at the hotel carrying countless new customers. The manager who knows a lot about math activates the intercom again and tells all the guests to go to the room with twice as many room as they are in: who’s in 1 goes to 2, who’s in 2 goes to 4, who’s in 3 goes to 6 and so on. By doing this, the manager frees up all the single rooms, which are infinite and which can thus accommodate the endless bus load of new customers. In this case, the director proved that 2 ℵ = ℵ.

This efficient hotel attracts great interest and after a short time the endless buses with countless number of customers start arriving. The director remembers that the primes (the natural numbers greater than 1 that contain only 1 and themselves as divisors) are infinite and that the new arrangements are based on this. All guests already in the hotel must go to the room opposite the number 2 (the first of the primes) raised to the power of the room number you are in. The guest must go at 9 to 29that is, to room 512.

The manager then turns to the customers of the first bus and says that each occupant 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. Next, the customer in seat 9 has to go to room 39, i.e. for 19683. The director assigns the next prime number to the second bus – 5 – with the same instructions for raising the force with respect to the occupied seat. The third bus will have 7 as the starting point, the fourth at 11, the fifth at 13 and so on to infinity, because the primes 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 manager work because they are closely related to the lowest possible level of infinity. This brings us back to the concept of the fundamental relationship of a countable: ℵ by Cantor.

To accommodate existing guests and new clients, the manager used only natural numbers. If he were to deal with an order greater than infinity, like the real numbers, then his strategy would no longer work because he would have no way of systematically understanding each number. The number of rooms in a hotel is infinite, but it is an infinite number: there are a number of rooms equal to the number of positive rooms that reach infinity. If a bus arrives with an infinite number of customers (none of whom are sitting in a numbered seat and all have a different name, so to speak), the manager will always end up with someone who is not counted and still needs a room. .

This paradox, which we have simplified a little, was invented by a German mathematician David Hilbert Specifically to show properties of the concept of infinity and to help understand the orders of infinity. Perhaps the clearest evidence of how difficult it is for our minds to perceive, which we face every day with limits, is that there can be different infinities, and above all that there are mathematical conditions that define their properties. It is no coincidence that the concept of infinity has astonished many philosophers and has led over the centuries to comparisons and disputes, more complex than this article.

Thelma Binder

"Explorer. Devoted travel specialist. Web expert. Organizer. Social media geek. Coffee enthusiast. Extreme troublemaker. Food trailblazer. Total bacon buff."

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