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What is infinity?

The scientific approach 26.04.2017 at 08:00

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Hotel Hilbert

Our young couple were absent for more than two months. Returning to the island, they immediately went to the Wizard.

— the return! — said the Wizard.

— So you want to know anything about infinity?

— you Have a good memory, ' replied Annabelle.

— well, Well — didn't argue a Wizard. — The first thing we need to do is carefully define our terms. What exactly do you mean by "infinite"?

— For me it means there is no end, — said Alexander.

— I would say the same thing, ' said Annabelle.

Is not quite satisfactory, ' said the Wizard. — The circle has no beginning and no end, and yet you would not say that it is infinite: it has only a finite length, although it contains an infinite number of points. I want to talk about infinity in the strict sense used by mathematicians. Of course, this word has other applications. For example, theologians often refer to the infinity of God, although some of them are honest enough to admit that in relation to God, this word is used not in the same sense as something else. I don't want to slight the theological or any other non-mathematical use of the word, but I want to make clear that the subject of our discussion is infinity in the pure mathematical sense of the term. And for him, we need a precise definition.

Obviously, the word "infinite" is an adjective, and above all, we must agree on what type of objects it is applicable. What kind of objects can be considered finite or infinite? In mathematical use of the term such objects are sets, or set of objects that can be finite or infinite. We say that a set of objects has a finite or infinite number of members, and now you need to make these concepts precise.

the Key role here is played by the concept of odnoosnogo matching between the two sets. For example, two sets — a flock of seven sheep and a grove of seven trees — are connected to each other as none of them is associated with a pile of five stones, because a lot of the seven sheep and a lot of the seven trees can be joined in pairs (for example, tied to each tree for the sheep) so that each sheep and each tree will belong to exactly one pair. In mathematical terminology this means that the set of seven sheep can be put in 1-1-digit according to the set of the seven trees. Another example. Let's say that, once in the auditorium, you see that all the seats are occupied, no one is and no one, no one sits on my lap, every place sits one and only one person. Then, not counting the number of people or the number of seats, you know that these numbers are equal, as many people are in 1-1-digit according to the set of places: each person corresponds to the place which he occupies.

I know that you are familiar with the set of natural numbers, although they may not know what it's called. Natural numbers are numbers 0, 1, 2, 3, 4... That is, a natural number is zero or any positive integer.

— is A non-natural number exists? asked Annabelle.

— No I've never heard of — smiled a Magician — and, I must confess, I find your question very funny. Anyway, from now on I will use the word number in the sense of natural number, unless something to the contrary. If given a positive integer n, what is then the assertion that a certain number has exactly n elements? For example, what does the assertion that my

right hand in exactly five fingers? This means that I can set 1-1-digit correspondence between the many fingers of my right hand with plenty of positive integers from 1 to 5, considering that the thumb is 1, index — 2, medium — 3, unnamed — 4, and pinky 5. In the General case for any positive integer n, the set contains (exactly) n elements, if it is possible to establish a 1-1-digit match between this variety and plenty of positive integers from 1 to n.

a set containing n elements, is called s p-element set. The process of establishing a 1-1-correspondence between the digit of an n-element set, and its many positive integers has a common name — account. Yes, this is the essence of the account. So, I explained to you what it means for a set to have n elements, where n is a positive integer. What if n = 0? [Newline]means for many to have 0 elements? Obviously, this means that the set has no elements.

— Such sets exist? — asked Alexander.

— There is only one such a lot, replied the Wizard. — It is called the empty set and is highly useful for mathematicians. Without him constantly would have to make exceptions, and everything would be very cumbersome. For example, we want to talk about the many people in the theater at the moment. It may happen that this point in time in the theatre in General, there are no people, and in this case we say that the set in the theatre's empty of people — just the same as talking about the empty theater. It should not be confused with the theatre at all! The theatre continues to exist as a theatre; it just may not be a single person. Similarly, the empty set exists as a set, but it has no elements.

I remember a wonderful occasion. Many years ago I was told about the empty lot cute lady musician. She was surprised and asked: "Mathematics really apply this concept?" I said, "Certainly applied".

She said, "Where?" "Everywhere," I replied. She pondered a while and said, "Oh, yeah. I guess it's like a musical pause." I think it was a very good analogy! One amusing incident connected with Smalliana. When he was a student at Princeton University, one of the famous mathematicians during the lecture said that he hated the empty set. In his next lecture, he used an empty set. Smullyan, raised his hand and said, "I thought you said you don't like the empty set". "I said I don't like the empty set, — replied the Professor. — I never said I don't use the empty set!"

— You haven't told us what you mean by "finite" and "infinite," said Annabelle. — Are you going to explain?

— this is What I do, — replied the Magician. Everything I told you before, leads to the definition of these terms. Many of course, if there is such a natural number n that the set contains exactly n elements, and this, as we remember, means that this set can be put in 1-1-correspondence with digit positive integers from 1 to n. If no such natural number n exists, then the set is called infinite. It is very simple. Thus, the 0-element set of course, the 1-element set of course, the 2-element set, of course ... and n-cell

many of course, where n is any natural number. But if for any natural numbers n is false that the set contains exactly n elements, then this set is infinite. So, if the set is infinite, then for any natural number n, if we remove from this set of n elements, there still remain elements — in fact, still remain an infinite number of elements.

— do You understand why this is true? Let us first consider a simple task. For example, I remove one element from an infinite set. What is left, is sure to be endless?

— have it so! said Annabelle.

— Exactly! — confirmed Alexander.

— Well, you're right, but can you prove it?

, Young people think, but the proof was a little difficult for them. Everything seemed too obvious to require proof. However, it is easy to prove from the definitions of the terms "finite" and "infinite". The data definition you want to apply for this. How do you prove it?

the Magician had a little push young people to the right decision, but in the end, they found evidence that gave it.

the Hotel Hilbert

— Infinite variety, — said Wizard, — possess some strange properties, sometimes referred to as paradoxical. In fact, they are not paradoxical, just slightly affect the first

acquaintance with them. This is well illustrated by the famous story of Hilbert's hotel. Take an ordinary hotel in which a finite number of rooms, say, one hundred. Let's say that all the rooms are occupied each by one tenant. A new person arrives and wants to rent a room for the night, but neither he nor one of the tenants of the hotel is not willing to share a room with another person. Then it is impossible to place the hotel of the new visitor as it is not possible to establish a 1-1-correspondence between the digit 101 man and 100 rooms.

However, with infinite hotel (if you can imagine such) the situation is different. In Hilbert's hotel infinite number of room: one for each positive integer. Rooms are numbered sequentially: room 1, room 2, room 3... n... and so on. One can imagine that the rooms are located in a linear fashion: they start at a certain point and continue to the right endlessly. Is the first number, but not the last! It is important to remember that there is no last number, just as there is no last natural number. Then again, it is assumed that all rooms are occupied each room for one person. There is a new visitor and wants to rent a room. It is interesting that now it can be placed in the hotel. Neither he nor one of the tenants of the hotel is not willing to share a room with another person, but each tenant of the hotel agreed to change my room to another if asked to do so.

How to report on a new tenant?

— Now let's move on to another task, — continued the Wizard after discussion, the solution of the previous problem. — Consider the same hotel. However, now instead of one person comes in an infinite number of new guests: one for each positive integer n. Let's call the old tenants of the hotel P1, P2... PN... and new visitors Q1 , Q2... Qn... Q All-persons wish to be placed in the hotel. What is unusual is that it is possible!

How to do it?

now consider the more interesting problem. Take an infinite number of hotels: one for each positive integer. The hotels are located on a rectangular area:

the Entire chain of hotels managed by one administration. All rooms in all hotels are busy. Once in order to save energy administration decides to close all the hotels, except one. However, it needs to accommodate all the occupants of all the hotels at the only hotel, one tenant in one room.

is it Possible?

— You see that reveal to us these challenges, ' continued the Wizard. They show that an infinite set has a strange property: it can be put in 1-1-digit line with a part of him. Let's define it more precisely.

the set a is called subset of b if every element of A is an element of V. for Example, if a is the set of integers from 1 to 100, In an array of numbers from 1 to 200, And there is a subset of V. If E — the set of even integers, and N is the set of all integers, E is a subset of N. a Subset of A set is called a proper subset if A is a subset of V but does not contain all of the elements of V. in Other words, A is a proper subset of b if A is a subset of V, but is not a subset of A. Let R is the set of all positive integers {1, 2, 3... n...}, R is the set of all positive integers without a unit {2, 3... n...}. In the first task about the hotel Hilbert we have seen that between R and R - can be installed 1-1-digit line, and yet R is a proper subset of R! Yes, an infinite set can have a strange property: it can be put in 1-1-digit according to its own subset! It has long been known. In 1638, Galileo showed that the squares of positive integers can be put in 1-1-digit line with those numbers.

it Seemed to be contrary to the ancient axiom that the whole is greater than any of its parts.

— is it not? — asked Alexander.

— actually there is no contradiction, ' replied the Wizard. — Suppose that A is a proper subset of V. Then one of the meanings of the word "more"— more A, namely in the sense that In contains all of the elements and those elements that are not in A. However, this does not mean that numerically exceeds A.

— I don't Think I understand, ' said Annabelle. — What do you mean by the term "numerically superior"?

— Good question! — said the Wizard. — First of all, what, in your opinion, I mean that A has the same magnitude as In?

— I guess that means between A and b can be set 1-1-digit line, ' replied Annabelle.

— Right! What, in your opinion, I mean, saying that A magnitude less or that the number of elements And less than the number of elements In?

— I guess that means that you can establish a 1-1-correspondence between the digit A and a proper subset of V.

— Nice try — approved Magician, but this version is not suitable. Such a definition would fit perfectly for finite sets. The trouble is that in some cases it is possible to establish a 1-1-correspondence between the digit A and a proper subset, but it is also possible to establish a 1-1-correspondence between digit V and a proper subset of A. In this case, you will be able to say that each of these sets is less than the other? For example, suppose that On — the set of odd numbers, and E — the set of even numbers. Obviously, between O and E can be set 1-1-digit match.

However, it is also possible to establish a 1-1-correspondence between a digit and a proper subset of E.

you can also establish a 1-1-correspondence between the character E and a proper subset of O.

Now you, of course, not to say that O and E have the same size, and still Of less than E, and E less than! No, this definition does not work.

— Then what is the definition of the relationship "...less than..." for sets? asked Annabelle.

the Correct definition is formulated as follows. But less than or more than A, if the following conditions exist: (1) it is possible to establish a 1-1-correspondence between the digit A and a proper subset; (2) it is impossible to establish a 1-1-correspondence between the digit and the entire set

to correctly say And less In need to satisfy both of these conditions. The assertion that A less In, means first of all that is possible to establish a 1-1-correspondence between the digit A and a subset of b, and that any 1-1-digit correspondence between a and the subset does not cover all elements of B.

now, there is a fundamental question. Any two infinite sets have the same size, or are there infinite sets of different sizes? This is the first question you have to answer in the theory of infinity, and, fortunately, answered by Georg Cantor in the late nineteenth century. The answer caused a storm and spawned a whole new branch of mathematics, the branches of which are just fantastic!

I will tell you the answer is Cantor at our next meeting. Yet think about it, what was the answer. Same largest all infinite sets, or some of them are different?

the Decision

1. First we will show that adding one element to a finite set a finite set is obtained. Let's say that many And of course. By definition this means that for some natural number n, the set a has n elements. If you add it And another item will be a set with n+1 elements, which, by definition, of course.

this immediately implies that the removal of an element from an infinite set, there must be an infinite set, because if it was finite, then, returning the removed element, we would get the original set, which would be the final, and the condition it is infinite.

2. The hotel only need to ask each of the guests to move one room to the right. In other words, the occupant of room 1 moves to room 2, the occupant of room 2 moves into room 3...

the tenant of room n moves into room n+1. Since this hotel does not have the last room (as opposed to the more normal end hotels), none of the guests will not be on the street. (At the end of the hotel, the occupant of the last room would be without a job.) After this move number 1 is freed, and new arrivals can take it.

Mathematically, in this case, you need to install 1-1-digit correspondence between the set of all positive integers with many positive integers beginning with 2. Of course, the hotel Manager could do the same with hundreds of millions of new visitors, if they came at the same time. He just asked every guest to move to one hundred million and one room to the right (the tenant of room 1 went to room 100000001, the tenant of room 2 in a room 100000002, and so on). For any natural number n the hotel was able to accept n new guests, moving the occupant of each room on the rooms to the right and thereby releasing the first n rooms for new guests.

3. If there comes an infinite number of new guests Q1 , Q2... Qn... need to act a little differently. One of the false solutions is the following. The Manager asks each of the old guests to move one room to the right and gives one of the visitors to the empty room 1. He then again asks everyone to move one room to the right and strikes the second guest in a vacant room 2.

Then this procedure is repeated again and again an infinite number of times, and sooner or later all the new guests move into the hotel.

Oh, what is troublesome is the solution!

no man takes a room permanently, and all guests cannot be accommodated for any finite period of time: requires an infinite number of displacements. No, everything can be settled with a single move. Can say what?

This movement is that each of the old guests doubles the number of my room, that is, the occupant of room 1 moves to room 2, the occupant of room 2 moves to room 4, the occupant of room 3 passes

to the number 6. the tenant of room n moves into room 2n. Of course, all this is done simultaneously, and after such a move all even numbers are occupied, and an infinite number of odd numbers freely. So, the first new guest in Q1 is 1, Q2 is in room 3, Q3 — 5 and so on (Qn goes to room 2n-1).

4. First "enumerate" all the guests all the rooms in all the hotels in accordance with the following plan:

so, each guest marked a positive integer. Then all asked to leave the room and wait a bit on the street. The administration covers all the hotels, except one, and asks each of the guests to occupy the hotel room, which he had designed: the guest from room n goes to room n.

the Passage from the book of Raymond Smullyan "Satan, Cantor and infinity and other puzzles"

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