Actual and potential infinity. How did infinity become a subject of exact science? Fractals and infinity

Actual infinity

Here's the first argument:

1. Actual infinity cannot exist.

2. A beginningless series of temporary events is actual infinity.

3. Consequently, a beginningless series of temporal events cannot exist.

Let's look at the first premise first: Actual infinity cannot exist.

What do I mean by actual infinity? A set of objects is considered to be actually infinite if Part of this set is equal to its to the whole. So for example, which rad is longer:

2,3,4,5,6,… or 0,1,2,3,4,5,6,…?

According to generally accepted mathematical representations, these series are equivalent because they are both actually infinite. This seems strange: after all, in the right row there are two numbers that are missing in the left. But this only shows that in an actually infinite set, the part (left row) is equal to the whole (right row).

For the same reason, mathematicians claim that the series of even numbers is equal to the series natural numbers- despite the fact that the series of all natural numbers contains all even numbers plus an infinite number of odd numbers.

1,2,3,4,5,6,7,8,…

There is no need to confuse concepts relevant infinity - and potential infinity.

According to the great German mathematician David Gilbert, the main difference between actual and potential infinity is this. The potentially infinite is always something increasing and having infinity as its limit, while actual infinity is a completed whole, actually containing an infinite number of objects.

An interesting example these two types of infinity can serve as two series of events: those that occurred before And after any point in the past.

Take, for example, the moment in 1845 when Georg Cantor, the father of set theory, was born.

In both cases we are referring to events that actually happened.

The point called “present time”, of course, does not stand still, but slides forward. (In essence, this is the boundary between events that have already been realized and those that have not yet been realized.) Therefore, the number of events "after"(i.e., between 1845 and the present), although at each specific moment the finite, is constantly increasing. It is never fully realized, and therefore potentially endlessly.

But the series of events “before” is fully realized, completed and does not increase. And if the atheists are right, and the Universe did not have a beginning, then such a series is endless. Infinite relevant, really.

In the course of our reasoning, it is very important not to confuse these two concepts (actual and potential infinity).

The second clarification concerns the word “exist.” When I say that actual infinity cannot exist, I mean exist in real world, or exist not only in the mind. I do not at all deny the legality of using the concept of actual infinity in mathematics(operating only with mental reality). I only assert that actual infinity cannot exist in physical world stars, planets, stones and people.

A few examples will show the absurdity of this assumption.

Let's assume that there is a library containing a truly infinite number of books. Let's imagine that the books in it are only two colors, black and red, and that they stand on the shelves, alternating: black, red, black, red, etc. If someone tells us that the number of black books is equal to the number of red ones, we probably won't be surprised. But will we believe it if they tell us that the number of black books is equal to the number of black and red books together? Indeed, in such a collection we will find all the black books plus an infinite number of red books!

Or let's imagine that we have books of three colors, four, five or even a hundred. Would we believe that there are as many books of the same color as there are total books in the library?

Or imagine that in the library infinite number of book colors. One might suppose that in an infinitely large library there would be one book for each of an infinite number of colors. But this is not necessarily the case. According to mathematicians, if the number of books is truly infinite, then for each of the infinite number of colors there can be infinite number of books. Thus we get an infinity of infinities! And yet, if we take all the books of all colors, there will be no more of them than books of only one color.

Let's continue our reasoning. Let's assume that each book has a number printed on its spine. Since the library is truly endless, every possible number printed on any of the books. Therefore, we cannot add another book to the library, because what number should we give it? All rooms are already occupied. Thus, a new book cannot be given a number! But this is absurd, since in reality objects can always be numbered.

If an infinite library existed, it would be impossible to add another book to it. (Is it because it would already include all the existing books, and there would simply be nowhere to get a new one? No, because it’s enough to tear out a leaf from each book of the first hundred, glue them together, put this new book on the shelf, and that’s it - a library replenished!) Therefore, the only possible conclusion suggests itself: a library that is actually infinite cannot exist.

But suppose we can add to this library, and I put the book on the shelf. According to mathematicians, the number of books in the library remains the same. How can it be? After all, my experience says: if I put a book on a shelf, then there are more books there, and if I take it down, then there is one less.

It's easy for me to imagine myself directing and filming this book. Should I really seriously believe that when I add books, their number does not increase, and when I remove them, their number does not decrease? What if I add to this library an infinite number or even an infinity of infinities of books? Is there really not a single more book in the library now than before? I find this hard to believe. And you?

Now let's do the opposite issue books from the library. Let's say we released book number eight on Monday. Hasn't the number of books decreased by one?

On Tuesday we will give out all books with odd numbers. An infinite number of books have gone, but mathematicians will say that there are no fewer books in the library.

Let's say that on Wednesday we gave out books numbered 4, 5, 6,... and ad infinitum. In one fell swoop, the library was almost completely empty, the infinite number of books was reduced to a finite number: three. But excuse me, this time we gave out as many books as on Tuesday! Why is there such a difference? And who would believe that such a library could actually exist?

All these examples illustrate the fact that actual infinity cannot take place in the physical world. I want to emphasize again: this does not threaten the theoretical system introduced into modern mathematics by G. Cantor. Moreover: even such enthusiasts of mathematical theories of the infinite as D. Gilbert readily agree that the concept of actual infinity is only idea, has nothing to do with the real world. Therefore, we have the right to conclude: actual infinity cannot exist.

Second parcel: A series of events in time that has no beginning represents actual infinity.

By "event" I mean any change that occurs in the physical world. That is: if a series of past events (or changes) always goes into the past and never has a beginning, then in this case, taken all together, these events constitute an actually infinite set.

Let's say we ask where such and such a star came from. We are told that it appeared as a result of the explosion of a star that existed before. Then we ask where it came from that star? She, too, arose from a star that existed before her. Where is this star from? From another, previous star - and so on. This series of stars will be an example of a series of events without beginning in time.

Then, if the Universe has always existed, the series of all past events in their totality will constitute actual infinity: because each event in the past was preceded by another event. Thus, the series of past events will be infinite.

But won't he potentially endless? No, for we have seen that the past is complete and actual - only the future can be characterized as potentially infinite. Therefore, it seems obvious that the beginningless series of events in time is actual infinity.

This brings us to the desired conclusion. a beginningless series of events in time cannot exist.(We established earlier that the actually infinite cannot exist in reality. And if a beginningless series of temporal events is actual infinity, then such a series cannot exist.)

This means that a series of all past events must have a beginning. But the history of the Universe is a series of all accomplished events! Therefore, the Universe must have a beginning.

A few examples will clarify this argument.

We know that if actual infinity could exist in reality, it would be impossible to add anything to it. But there are additions to the series of events in time every day - or at least it seems so to us. If this series is actually infinite, then the number of events that have happened up to the present moment is no more than. say, the number of events before 1789 or any other point in the past, however distant.

Another example. Let us imagine that two planets have been revolving around the Sun for an eternity. Let’s assume that one completes its orbit in three years, the other in a year. Thus, for every revolution of one there are three revolutions of the other. Question: if they move forever, which of these planets made more orbital revolutions? Answer: both made the same number of revolutions. But this is clearly absurd, because common sense suggests: the longer they rotate, the more the gap increases. How can the number of revolutions be equal?

Or, finally, let's say that we met an alien. He claims that he has been counting for ages, and now he finishes: ... 5, 4, 3, 2, 1, 0. But we can ask: why didn’t he finish counting yesterday7 or even a year ago? Did he really not have enough time? How so? After all, an infinite number of years had passed before last year, which means he had enough time. What happens? No matter how far into the past we go, we will never catch him counting. Therefore, it cannot be true that he has been doing this for all eternity.

These examples highlight the absurdity of the idea of ​​a beginningless series of events in time. Since such a series is actually infinite, and actual infinity cannot exist, then this series is impossible. It means that The universe once began to exist, Q.E.D.

Our young couple had been gone for over two months. Returning to the island, they immediately went to the Wizard.

- Welcome back! - said the Wizard.

- So you want to know something about infinity?

“You have a good memory,” Annabelle answered.

“Well, good,” the Wizard did not object. “The first thing we need to do is to carefully define our terms.” What exactly do you understand by the word “infinite”?

“For me it means no end,” said Alexander.

“I would say the same,” Annabelle confirmed.

“This is not entirely satisfactory,” said the Wizard. - A circle has neither beginning nor 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 precise sense used by mathematicians. Of course, there are other uses for this word. For example, theologians often refer to the infinity of God, although some of them are honest enough to admit that this word is not applied to God in the same sense as to something else. I do not wish to disparage the theological or any other non-mathematical uses of the word, but I do want to make it clear that the subject of our discussion is infinity in the purely mathematical sense of the term. And for this we need a precise definition.

Obviously, the word "infinite" is an adjective, and first of all we must agree on what sort of objects it applies to. What kinds of objects can be considered finite or infinite? In the mathematical application of the term, such objects are sets, or collections of objects, which can be finite or infinite. We say that a set of objects has a finite or infinite number of members, and now we need to make these concepts precise.

Key role The concept of one-to-one correspondence between two sets comes into play here. For example, two sets - a flock of seven sheep and a grove of seven trees - are related to each other in a way that neither of them is related to a pile of five stones, because a set of seven sheep and a set of seven trees can be connected in pairs (for example , tying a sheep to each tree) 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 into a 1-1 digit correspondence with the set of seven trees. Another example. Let's say that when you get into a theater hall, you see that all the seats are occupied, no one is standing and no one is sitting on anyone's lap, one and only one person is sitting in each seat. Then, without counting the number of people or the number of places, you know that these numbers are equal, since the set of people is in 1-1 digit correspondence with the set of places: each person corresponds to the place he occupies.

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

- Does an unnatural number exist? - Annabelle asked.

“No, I’ve never heard of this,” the Wizard grinned, “and I must admit, I find your question very funny.” Be that as it may, from now on I will use the word number in the sense of a natural number, unless otherwise stated. Given a natural number n, what does it mean to say that a certain set has exactly n elements? For example, what does it mean to say that on my
Are there exactly five fingers on your right hand? This means that I can establish a 1-1 digit correspondence between many fingers of my right hand with a set of positive integers from 1 to 5, assuming that thumb corresponds to 1, index - 2, middle - 3, ring - 4 and little finger - 5. In general, for any positive integer n, a set contains (exactly) n elements if a 1-1-digit correspondence can be established between this set and a set of positive integers from 1 to n.

A set containing n elements is also called an n-element set. The process of establishing a 1-1-digit correspondence between an n-element set and a set of positive integers has a common name - counting. Yes, this is exactly the essence of the account. So I've explained to you what it means for a set to have n elements, where n is a positive integer. What if n = 0? What
does it mean for a set to have 0 elements? Obviously, this means that the set has no elements at all.

— Do such sets exist? - asked Alexander.

“There is only one such set,” answered the Wizard. - It is called the empty set and is extremely useful for mathematicians. Without it, exceptions would constantly have to be made, and everything would become very cumbersome. For example, we want to talk about a lot of people in a theater in this moment. It may happen that at this moment in time there are no people in the theater at all, and in this case we say that the set of people in the theater is empty - just as we talk about an empty theater. It should not be confused with theater in general! The theater continues to exist as a theatre; there just may not be a single person in it. Similarly, the empty set exists as a set, but it has no elements.

I remember a wonderful incident. Many years ago I talked about the empty set to a nice lady musician. She was surprised and asked: “Do mathematicians really use this concept?” I replied: “Of course they do.”
She asked: “Where?” “Everywhere,” I replied. She thought for a moment and said, “Oh, yes. I guess it's like musical breaks." I thought that was a very good analogy! One funny incident involves Smullyan. When he was a student at Princeton University, one of the famous mathematicians said during a lecture that he hated the empty set. In his next lecture he used the empty set. Smullian raised his hand and said, “I thought you said you didn’t like the empty set.” “I said that I don’t like the empty set,” the professor replied. “I never said I didn’t use the empty set!”

“You haven’t yet told us what you mean by ‘finite’ and ‘infinite,’” said Annabelle. -Are you going to explain?

“That’s exactly what I’m getting at,” answered the Wizard. “Everything I told you before leads to the definition of these terms.” A set is finite if there is a natural number n such that this set contains exactly n elements, and this, as we remember, means that this set can be put into a 1-1-digit correspondence with positive integers from 1 to n. If such a natural number n does not exist, then the set is called infinite. It's very simple. Thus, a 0-element set is finite, a 1-element set is finite, a 2-element set is finite... and an n-element
the set is finite, where n is any natural number. But if for any natural number n it is false that the set contains exactly n elements, then the set is infinite. This means that if the set is infinite, then for any natural number n, if you remove n elements from a given set, there will still be elements left in it - in fact, there will still be an infinite number of elements left.

- Do you understand why what was said is true? Let's look first simple task. Let's say I removed one element from an infinite set. What remains will necessarily be endless?

- It seems so! - Annabelle said.

- Exactly! - Alexander confirmed.

- Okay, you're right, but can you prove it?

The young people thought about it, but the proof turned out to be difficult for them. Everything seemed too obvious to require proof. However, this is easy to prove from the very definitions of the terms “finite” and “infinite”. These definitions must be applied for this. How to prove this?

The wizard had to push the young people a little towards the right solution, but in the end they found a proof that suited him.

“Infinite sets,” said the Wizard, “have some strange properties, which are sometimes called paradoxical.” In fact, they are not paradoxical, they are just slightly surprising at first
getting to know them. This is well illustrated by the famous story of Gilbert's Hotel. Let's take an ordinary hotel with a finite number of rooms, say one hundred. Let's assume that all the rooms are occupied and there is only one occupant in each of them. arrives new person and wants to rent a room for the night, but neither he nor any of the hotel residents want to share their room with another person. It is then impossible to accommodate a new visitor in the hotel, since it is impossible to establish a 1-1 digit correspondence between 101 people and 100 rooms. However, with an infinite hotel (if you can imagine one) the situation is different. Hilbert's hotel has an infinite number of rooms: one for each positive integer. The rooms are numbered sequentially: number 1, number 2, number 3... number n... and so on. You can imagine hotel rooms arranged in a linear order: they start at a certain point and continue to the right ad infinitum. There is the first number, but not the last! It is important to remember that no last issue, just like there is no last natural number. Further, it is again assumed that all rooms are occupied: there is one person in each room. A new visitor appears and wants to rent a room. Interestingly, it can now be placed in a hotel. Neither he nor any of the hotel residents want to share his room with another person, but each hotel resident agrees to change his room for another if asked.

“Now let’s move on to another problem,” the Wizard continued after discussing the solution to the previous problem. — Let's consider the same hotel. However, now, instead of one person, an infinite number of new guests arrive: one for each positive integer n. Let's call the old residents of the hotel P1, P2... Pn... and the new arrivals Q1, Q2... Qn... All Q-persons want to be accommodated in the hotel. What's extraordinary is that this is possible!

How to do it?

Now let's look at an even more interesting problem. Let's take an infinite number of hotels: one for each positive integer. The hotels are located on a rectangular square:

The entire chain of hotels is managed by one administration. All rooms in all hotels are occupied. One day, in order to save energy, the administration decides to close all hotels except one. However, to do this, you need to place all the residents of all hotels in a single hotel - one resident in one room.

Is it possible?

“You see what these tasks reveal to us,” continued the Wizard. “They show that an infinite set has a strange property: it can be put into a 1-1-valued correspondence with its own part.” Let's define this more precisely.

A set A is called a subset of a set B if every element of A is an element of B. For example, if A is the set of numbers from 1 to 100, B is the set of numbers from 1 to 200, then A is a subset of B. If E is the set of even numbers, and N is the set of all numbers, then E is a subset of N. A subset A of a set B is called a proper subset of B if A is a subset of B, but does not contain all the elements of B. In other words, A is a proper subset of B if A is a subset of B, but B is not a subset of A. Let P be the set of all positive integers (1, 2, 3... n...), P- be the set of all positive integers without one (2, 3... n. ..). In the first problem about Hilbert's hotel, we saw that a 1-1-digit correspondence can be established between P and P-, and yet P- is a proper subset of P! Yes, an infinite set can have the strange property that it can be put into a 1-1 digit correspondence with its own subset! This has been known for a long time. In 1638, Galileo showed that the squares of positive integers can be put into a 1-1 digit correspondence with the numbers themselves.

This seemed to contradict the ancient axiom that the whole is greater than any of its parts.

- Is not it so? - asked Alexander.

“In fact, there is no contradiction,” answered the Wizard. — Let us assume that A is a proper subset of B. Then in one of the senses of the word “more”— B is greater than A, namely in the sense that B contains all the elements of A and also those elements that are not in A. However, this does not mean that B outnumbers A.

“I don’t think I understood,” Annabelle said. —What do you mean by “outnumbered”?

— Good question! - said the Wizard. - First of all, what do you think I mean when I say that A has the same magnitude as B?

“I guess that means there's a 1-1 digit correspondence between A and B,” Annabelle replied.

- Right! What do you think I mean when I say that A is smaller in size than B, or that the number of elements of A less number elements B?

“I suppose this means that it is possible to establish a 1-1 value correspondence between A and its own subset B.”

“Not a bad try,” the Wizard approved, “but this version is not suitable.” This definition would work perfectly for finite sets. The trouble is that in some cases you can establish a 1-1-valued correspondence between A and a proper subset of B, and you can also establish a 1-1-valued correspondence between B and a proper subset of A. In this case, you can say that each of are these sets smaller than the other? For example, let O be the set of odd numbers, and E be the set of even numbers. Obviously, a 1-1-digit correspondence can be established between O and E.

However, it is also possible to establish a 1-1-valued correspondence between O and its own subset E.

You can also establish a 1-1-digit correspondence between E and its own subset O.

Now, of course, you will not say that O and E have the same magnitude, and yet O is less than E, and E is less than O! No, this definition does not work.

- Then what definition of the relation “...less than...” is suitable for sets? - Annabelle asked.

The correct definition is formulated as follows. A is less than B, or B is greater than A, if the following conditions are satisfied: (1) a 1-1-valued correspondence can be established between A and a proper subset of B; (2) it is impossible to establish a 1-1-digit correspondence between A and the entire set B.

To correctly say that A is less than B, both of these conditions must be met. The statement that A is less than B means first of all that it is possible to establish a 1-1-valued correspondence between A and a subset of B, and also that any 1-1-valued correspondence between A and a subset of B does not exhaust all the elements of B.

Now a fundamental question arises. Do any two infinite sets have the same size, or are there infinite sets of different sizes? This is the first question that needs to be answered when constructing the theory of infinity, and, fortunately, Georg Cantor answered it at the end of the century before last. The answer caused a storm and gave birth to a whole new direction in mathematics, the branches of which are simply fantastic!

I will let you know Cantor's response at our next meeting. For now, think for yourself what this answer was. Are all infinite sets the same in size, or are there different ones among them?

Solutions

1. First we show that adding one element to a finite set produces a finite set. Let us assume that the set A is finite. By definition, this means that for some natural number n, the set A has n elements. If you add one more element to A, you get a set with n+1 elements, which by definition is finite.

It immediately follows from this that as a result of removing an element from an infinite set B, an infinite set should be obtained, for if it were finite, then by returning the removed element, we would obtain the original set B, which would be finite, and by condition it endlessly.

2. The hotel administration just needs to ask each of the guests to move one room to the right. In other words, the occupant of number 1 goes to number 2, the occupant of number 2 goes to number 3...
the inhabitant of number n goes to number n+1. Since this hotel does not have a last room (unlike more normal end hotels), none of the guests will end up on the street. (In the final hotel, the occupant of the last room would be without a seat.) After such a move, room 1 is freed, and the new arrival can take it.

Mathematically in in this case you need to establish a 1-1 digit correspondence between the set of all positive integers and the set of positive integers starting with 2. Of course, a hotel manager could do the same with a hundred million new guests if they arrived at the same time. He would simply ask each guest to move one hundred million and one room to the right (tenant number 1 would move to room 100000001, resident number 2 to room 100000002, and so on). For any positive integer n, the hotel could accept n new guests by moving the occupant of each room n rooms to the right, thereby freeing the first n rooms for new guests.

3. If an infinite number of new guests arrive Q1, Q2... Qn... you need to act a little differently. One of the false solutions is as follows. The manager asks each of the old guests to move one room to the right and moves one of the newcomers into the empty room 1. Then he again asks each of the old guests to move one room to the right and moves the second guest into the empty 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 a troublesome decision this is!

No one person occupies a room permanently, and it is impossible to accommodate all the guests in any finite amount of time: an infinite number of movements are required. No, everything can be settled with a single move. Can you tell which one?

This transfer consists of each of the old guests doubling the number of his room, that is, the occupant of number 1 goes to room 2, the occupant of number 2 goes to room 4, the occupant of number 3 goes
to room 6... the occupant of room n moves to room 2n. Of course, all this is done at the same time, and after such a move, all even numbers are occupied, and an infinite number of odd numbers are free. So the first new guest Q1 goes to number 1, Q2 goes to number 3, Q3 goes to number 5 and so on (Qn goes to number 2n-1).

4. First, we will “number” all guests of all rooms in all hotels in accordance with the following plan:

So, each guest is “marked” with a positive integer. Then everyone is asked to leave their rooms and wait outside for a while. After this, the administration closes all hotels except one, and asks each of the guests to take the hotel room that was assigned to him: the guest with number n goes to room n.

Philosopher a concept reflecting the boundlessness and limitlessness of the development of matter, the inexhaustibility of its knowledge. The place occupied by the concept of B. in the system of dialectical categories. materialism is determined by B.’s connection with such fundamentals. categories such as matter, motion, space and time. B. is the most total quantities. characteristic of moving matter. Considered with this most common quantities. On the other hand, the material world appears as infinite in space, infinite in time and infinitely diverse in its properties, in those specific forms, in which matter moves. Moreover, if B. in space and time describes the world as a whole, then B. of properties characterizes not only the world as a whole, but also each department. material object. Logically B. material world(in all its three aspects) can be considered as a consequence of the substantial nature of matter. Since matter is recognized as unity. a substance that is a causa sui (its own cause), to the extent that its development and movement cannot be limited by anything. If matter “opposes us as something given, as something uncreated and indestructible, then it follows that movement is also uncreated and indestructible” (F. Engels, Dialectics of Nature, 1955, p. 45). Natural the condition for such movement of matter is the existence of matter in time. On the other hand, being unity. the substance of the world, matter can be limited in its extent only by itself, which serves as an expression of its B. in space. It is obvious that the movement and development of matter, existing infinitely in infinite space, is carried out as unlimited. interaction of material bodies with each other. This infinite variety and number of connections between objects determines the inexhaustibility of the properties of both the whole world and any material object. Thus, the B. of the material world includes three aspects and cannot be considered in isolation from them, for these three aspects themselves, the three sides of the B. of the Universe, are closely related to each other. The concept of "B. world" plays big role in cosmology. Around the problem of B. there has always been and continues to be a struggle between idealism and materialism. Already Archytas of Tarentum (5th century BC) proved the B. of the world in space, interpreting the B. of the world as the possibility of continuous counting of distances in it up to B. However, such an idea of ​​the universe of the world leads to insoluble contradictions, called cosmological paradoxes. Thus, from the idea of ​​an infinite Universe uniformly filled with luminous stars, it necessarily follows that the firmament must be dazzlingly bright (Olbers' paradox). The attraction is endless large mass an infinite universe must lead to an infinite high speeds and mass accelerations (Zeliger's paradox). Both of these sharply contradict observations. Attempts to extend the second law of thermodynamics to the infinite Universe led to the conclusion about the beginning of the history of the Universe. The same conclusion was reached when the explanation of the phenomenon of red shift in the spectra of distant galaxies by their real “scattering” was extended to the entire Universe. These cosmological paradoxes led a number of scientists to abandon the idea of ​​the B. Universe in space and time, to attempts to “scientifically” substantiate the finitude of the world, which was essentially direct support for fideism. Cosmological Paradoxes, apparently, are the result of the extension to the entire Universe of laws established only for its finite regions. The specificity of such an object of study as the “Universe as a whole” explains the fact that we cannot extend to it not only the laws established for finite regions of the Universe, but even those established for arbitrarily large (but limited) regions of it. However, the reality of the world, like its materiality, must be confirmed by the entire course of the history of philosophy and science. A positive contribution to the solution of cosmological. problems, and thereby in revealing the specific nature of physical. B. of the world are different cosmological. models. In fact, cosmological. models represent an extrapolation of patterns, the validity of which has been confirmed for the region of the Universe known to us, to extremely large regions of the Universe (in the limit - to the entire infinite Universe). The possibility of such extrapolation finds some justification in the fact that such physical laws, such as the laws of conservation of mass, charge, energy, momentum, etc., which are essentially natural sciences. the expression of the laws of conservation of matter and motion, apparently, can be extended without restrictions to the Universe as a whole. However, it is necessary to constantly keep in mind the limitations of the cosmological method. models, since in this case the transition to B. of one or more characteristics of cosmological is considered. models, which are only approximate reflections of the infinite variety of properties of the real Universe. The first cosmological Newton considered the model. Considering space to be infinite, he came to the conclusion that in order for gravitating matter in infinite space not to gather into one gigantic mass, it must be evenly distributed in infinite space. However, this model leads to photometric and gravitational paradoxes, to eliminate which classical. cosmology failed. A significant step forward was the creation by Einstein of the general theory of relativity (1918), which connected geometric. properties of space and time (their metrics) with the distribution of gravitating masses. Einstein's attempts to obtain a solution to the equations of the general theory of relativity for the infinite Universe and thereby study its space-time structure were unsuccessful (Einstein assumed that the metric does not change with time). On this basis, Einstein and later De Sitter came to the conclusion that the world is spatially closed: a ray of light emitted by some source will return in a finite time to a fairly close vicinity of the source. Thus was born the “proof of the finitude of the world” by the general theory of relativity. Sov. scientist A. A. Friedman (1922) showed that from Einstein’s equations it is possible to obtain fundamentally different solutions, namely, solutions corresponding to an infinite world, if only we abandon Einstein’s assumption that the metric is unchanged over time. Friedman's model of the world must expand. This was confirmed by the discovery (1927) of a red shift in the spectra of distant galaxies. Subsequent studies showed that the geometry of the part of the Universe known to us is mainly determined by the average density of matter in it. At a density of 10-28, 10-29 g/cm 3 the space of the world is infinite, and the geometry of the world is Euclidean. A higher density corresponds to the closedness of space (Riemann geometry in the narrow sense of the word), and a lower density corresponds to openness (Lobachevsky geometry). In the first case, the world turns out to be spatially limited, in the second – infinite. However, the accuracy of average density determinations is so low that enormous errors are possible. Therefore, the question about the metric of the region of the Universe known to us remains open. But even if further research leads to the idea of ​​​​closedness (in the sense indicated above) of our region of the world (if the value of the average density turns out to be high enough), this will in no way give the right to judge the geometry outside the region of the Universe known to us. The conclusion about the spatial closedness of the region of the world known to us will essentially mean an approach to such regions and scales, which are characterized by different properties and measures than the types of matter and forms of its movement known to us so far. Thus, modern models are a reflection of the most important features of the region of the Universe known to us - the Metagalaxy. But they cannot in any way be considered models of the Universe as a whole. The other pole of the question about the B. of the material world in space and time is the problem of “infinitely” small distances and time intervals. Pointing to the B. of matter “deeply”, to the inexhaustibility of properties elementary particles, studied by modern microphysics, dialectic. materialism also associates with this a change in spatio-temporal relations in the microworld, since these relations are determined by the properties of material objects. In this regard big interest represents the hypothesis about the discontinuity of space and time in the microcosm, the so-called. "quantization" of space and time. In modern quantum field theory, where particles are considered as point formations, whole line characteristics (energy, mass, charge) of particles receives infinite values, which is devoid of physical meaning and indicates organic. flaw of theory. Currently time through a series of mathematical operations have succeeded in isolating endless expressions, but this is only a formal solution to difficulties that continue to exist in hidden form. An analysis of the difficulties with infinities shows that they are a consequence of the assumption of the existence of arbitrarily small wavelengths and arbitrarily small distances. As a result, there has been a tendency to overcome difficulties by introducing into the theory a certain minimum length and a certain minimum period of time, which should reflect the discontinuous nature of the space-time properties of matter in the microcosm. The term “quantization” of space and time itself is not precise. It evokes the idea of ​​certain elementary, very small parts, “bricks”, from which space and time are composed. In fact, the real meaning of the hypothesis of discontinuity of space and time is that in the microworld there is a definition. a boundary beyond which the spatio-temporal properties of material objects change radically. This change apparently consists in the fact that in microregions below the definition. limit, new world constants appear: the minimum extent of material objects and the minimum duration of the processes of change in these objects. Thus, the data of physics, as well as the data of cosmology, confirm and justify the correctness of dialectical-materialistic. B.'s understanding of the material world in space and time, which is based on the recognition of not only continuity, but also discontinuity, structure, and qualities. variety of space-time properties of moving matter. Above, the question of B. was considered from an ontological perspective. t.zr. But you can approach this issue from an epistemological perspective. t.zr. and consider the problem of the cognition of B. Obviously, B. is not an object of feelings. knowledge, it is not given directly in experience. Almost feelings. Human activity takes place entirely in the world of finite things and processes. Man directly perceives and studies only the finite. Knowledge of the infinite comes as a result of knowledge that has risen to the level of abstraction. The possibility of knowing B. is determined by the fact that B. is not next to the finite, but in unity with it. The unity of the finite and the infinite is the basis for the knowledge of B. Therefore, dialectical. materialism denies the unknowability of B.: "Everything true knowledge nature is the knowledge of the eternal, the infinite, and therefore it is essentially absolute" (Engels F., Dialectics of Nature, 1955, p. 186). In the progressive movement of thinking humanity, the contradiction between the absolute nature of human thinking and its implementation in individual limited ones is resolved thinking people, between the always limited object of human knowledge and B. of knowable nature “Therefore, the knowledge of the infinite is surrounded by two kinds of difficulties and can, by its very nature, be accomplished only in the form of some infinite asymptotic progress. And this is quite enough for us so that we have the right to say: the infinite is as knowable as it is unknowable, and this is all we need" (ibid.). Engels argued that every step of knowledge deepens our knowledge of material reality, but that no number of these steps exhausts logic. The question of knowledge of logic also has a formal-logical side that it allows (at least formally) to extend the action of laws found in a finite number of observations over a finite number of objects. , to an infinite number of cases of this type, to infinite areas. The extension of the results obtained to B. gives the established laws a form of universality and, thereby, necessity. But here the question arises about the admissibility of extrapolation to B., about the informal justification of the universality and necessity of the judgments derived. from experience. This question is all the more important because it is precisely such judgments that constitute the main content of science. But Kant could not solve it. Its strictly formalistic. attitudes ultimately led to the denial of the universality and necessity of judgments derived from experience; extrapolation to B. was declared illegal. Modern bourgeois The philosophy of the positivist direction generally refuses to recognize the universality and necessity of any judgments, believing that the extension of their action to an infinite number of cases cannot be logically justified and is an arbitrary act. And indeed, the question cannot be resolved within the framework of any formal logic. structures. Dialectical materialism claims that the problem of justifying the universality and necessity of judgments is not formally logical. problem. The universality and necessity of these judgments is justified by the practice of mankind. Philosophers, logicians, mathematicians, physicists, and astronomers have constantly turned to the study of B. for thousands of years. Materialism and idealism, dialectics and metaphysics, confidence in the knowability of the infinite world and agnosticism constantly collided and fought over different understandings of B.V. ancient philosophy in accordance with the general, weakly differentiated nature of science, ideas about philosophy were woven into one complex knot, from which it is difficult to isolate the actual philosopher. thread of reasoning. In the apeiron of Anaximander B. – main. characteristic of primary matter - appears as something highly indefinite, qualityless and therefore boundless and limitless. This vagueness and vagueness in B.'s understanding was gradually overcome by science; already Plato and Aristotle carried out logical. study B. The origins of Plato's concept of the infinite lead to the Pythagoreans. According to Plato, limit and infinity, “grown together,” are the principles contained in the “eternally existing” (“Philebus”). The infinite is that which can increase or decrease without limit. The nature of the infinite is that it is a “continuous movement forward.” In this capacity, the infinite cannot be known, because the infinite number of things and their signs makes our thinking about them indefinite. The task of cognition is not to fix thought on the limit or the infinite, but to find “intermediate members”, “quantitative certainty” of properties and relationships. In Plato's reasoning, B. acts as an unlimited possibility. increase or decrease. However, the recognition of the incomprehensibility of such an infinite leads to the search for a definition. relationships that could connect the limit and the infinite. If Plato's analysis of B. was auxiliary. means in its ethical research and was based on idealistic. character of the entire system, then for Aristotle the problem of B. was in close connection with the consideration of causality, space, time, motion ("Metaphysics", XI 10; "Physics", III 4–8). Not infinite as a symbol eternal kingdom Aristotle was interested in ideas, but the infinite in the real world. This approach to the problem reflected materialism. trends in Aristotle's philosophy. Aristotle formulated the subject of research this way: “Does the infinite exist and what is it?” And he answered: “In in a certain respect the infinite exists, but in another respect it does not. The infinite does not exist actually, like an infinite body or quantity perceived by the senses; the infinite is not self-sufficient. the beginning of being, as the Pythagoreans and Plato argued; the infinite cannot exist separately from the senses. items. The Infinite exists potentially; the infinite is movement - it “always becomes different and different.” Aristotle rejected the existence of an infinite series of causes, goals, and sources of movement. Aristotle made an attempt to classify the types of infinity, which consisted of indicating seven different cases of use of the word "B." in Greek language. Aristotle and Plato saw the characteristic and fundamental. B.'s feature is unlimited. quantity changes ( potential infinity ) and denied the existence of the infinite (current B.). In medieval philosophy two lines can be distinguished. On the one hand, purely theological. works limited to consideration of B. as an attribute of God; on the other hand, scholastic. controversy surrounding the issue of infinite divisibility. In both cases, skillful juggling of concepts and sophisticated logic. formalism essentially absorbed the very content of the question. But already in the 15th century. the process of breaking scholasticism began. At first, this process was expressed in a transition to the position of pantheism. Fusion of infinite deities. beings with infinite nature - ch. feature in the philosophy of Nicholas of Cusa, whom he considered his main. the task is to clarify the nature of “maximality” (“On learned ignorance”, 1410, ed., see Russian translation, 1937). The maximum is something greater than which nothing can exist; the maximum is identical with unity and being. The maximum is the infinite truth, which we “comprehend incomprehensibly.” The infinite cannot be known, for all knowledge consists in establishing relationships, proportions, and the infinite eludes all proportions: finito od infinitum nulla est proportia. Although the entire presentation of Nicholas of Cusa is fundamentally theological, God plays the role of a background, not a main one. the core of inferences. The dialectic of the infinite, developed by Nikolai Kuzansky, attracts attention. In B. the maximum and minimum coincide, opposites merge; an infinite straight line coincides with an infinite circle, an infinite triangle, etc. These dialectical ideas were perceived and developed by Giordano Bruno on a basis that was more distant from theology. It is characteristic that Bruno, as was generally typical of materialists, explored not just the concept of B., but the infinite Universe (“On the Cause, the Beginning and the One,” 1584, see Russian translation, 1934). This gave his work an anti-Church quality. direction. Taking the B. Universe as a starting point, Bruno analyzed the properties of such a Universe and considered it to be unified (since the infinite has no parts), motionless (since the infinite has nowhere to move), eternal (since the infinite is not born and does not dies). The existence of the infinite Universe contains all contradictions “in unity and harmony.” In B. there is no distinction between part and whole, therefore in eternity an hour is equal to a day, a day is equal to a year and, therefore, there are no more infinite hours than infinite centuries. Finally, in B., possibility does not differ from reality. The dialectic of the infinite in Bruno (as well as in Nicholas of Cusa) often bears the stamp of scholasticism. Bruno especially emphasized the unity, simplicity, and immobility of the Universe, without noting that to the same extent the infinite Universe is characterized by diversity, complexity, and movement. In the history of modern philosophy, the analysis of the concept of B. begins with Descartes’ refusal to enter into disputes about the infinite. This is motivated by the fact that the finite mind of man is not able to comprehend the infinite power of God. Therefore, where we do not see boundaries, we should talk about the indefinite (indefini), and not about the infinite (infini), which is inherent only in God. Formally maintaining loyalty to theology, Descartes, however, definitely expressed the idea of ​​the B. of an extended material substance, that is, of the B. of the material world. This opinion was supported and argued by Spinoza. B. substances Spinoza based on abs. the nature of its existence. The Infinite is an attribute of the absolute. Neither time, nor number, nor measure can be infinite, because... they will only help. means of imagination; infinite extent and duration. In his classification of the infinite (Letter to Meyer, April 20, 1663), Spinoza gave a definition of the concept of infinity that was close to that subsequently developed by Hegel. However, Spinoza did not single out this understanding of B. as true. For him, B. of an incomplete series was no less true. Basic Spinoza's thought was that the problem of B. can be solved only when they begin to draw a clear line between various types infinite, because the properties of these types are different. Much attention B.'s problem was given attention in English. materialistic philosophy of the 17th–18th centuries. Hobbes denied the existence of infinite numbers. Based on his understanding of space as “imaginary space,” he rejected the Cartesian idea of ​​the infinite extension of material substance (“Fundamentals of Philosophy,” parts 1–3, 1642–58, see Russian translation, 1926). Locke took the same position. Materiality is not identical to extension. The Universe as a collection of material bodies is finite and immersed in infinite empty space (vacuum). The concepts of finite and infinite were considered by Locke as modes of quantity (An Essay on Human Reason, 1690, see Russian translation, 1898). B. can only be attributed to space, duration and number. A person perceives only finite things, and the idea of ​​B. is created in the soul from the ability of unlimited repetition of concepts. quantities. Infinite means movement beyond any boundaries. The extreme boundaries of space, duration and number are beyond the "beyond the understanding of the soul." Toland sharply criticized Locke. Based on the recognition of self-motion, intrinsic. activity of matter, he denied the existence of empty space and reduced the question of the B. of space to the question of the B. of matter. Matter is infinite, because... “it is impossible to imagine” its boundaries (“Letters to Serena,” 1704, see Russian translation, 1927). Considering the properties of infinite matter, Toland largely repeated Bruno's ideas, but emphasized not the immobility of the infinite, but the infinite diversity and eternity of the development and movement of matter. Like all his predecessors, Toland denied the existence of endlessly relevant large numbers. He pointed out the need to distinguish between what is truly infinite and what can only be continued without end. With idealistic positions, Locke's concept of B. was criticized by Leibniz, who argued ("New Experiments on the Human Mind", 1700–05, ed. 1756, see Russian translation 1936) that the source of the idea of ​​B. lies within man and has deities nature. True B. lies in the absolute and is expressed in the necessity of God's existence. Franz. materialists of the 18th century They considered the position about the B. of nature in space and time indisputable, but understood it metaphysically - as an endless repetition of the same objects in an “empty”, identical space and time everywhere. Kant also did not go beyond the limits of metaphysics. For him, every B. is transcendental, i.e. it can never be considered as given; it only means what follows. the synthesis of units when measuring a quantity can never be completed ("Critique pure reason ", 1781, see Russian translation 1915). The fact that the unconditional, the infinite is taken as given, as an object and is placed in a series of conditioned and finite phenomena, and is, according to Kant, the cause of antinomies. The fact that a line can be continued to B. or that a series of changes can continue forever, assumes that time and space depend only on contemplation, since it in itself is not limited by anything. Thus, the B. series is based on the a priori nature of space-time forms. Hegel rightly reproached Kant for. subjectivism for such an understanding of infinite progress. But Kant only logically completed the views of his predecessors, who saw the basis of B. in the ability of reason to repeat any quantities. Kant took the next step, “hiding in the human head” not only B., but. and space and time itself. Hegel's understanding of B. (with its positive and negative sides) represents a natural consequence of the previous philosophical ideas. the opposite of finite; the infinite is not given in reality, it is a process, a movement, a continuous increase or decrease of something. quantities; the infinite is a simple repetition of the same things and processes; it is unknowable and cannot be grasped by the finite mind. Hegel's merit is that he is a man who is convinced of the power. mind, fell upon these metaphysics. and agnostic. views. Hegel showed that B. can be understood only in unity with the finite, that any finite contains the infinite, that the infinite is realized in the finite. Dialectical the unity of the finite and B. serves as the bridge along which humanity moves from the knowledge of the finite to the knowledge of B. Hegel ridiculed the romantic. admiration and metaphysical horror of B. From Hegel's point of view, the finite is not truly existing. Although from the side of its being the finite is real, the truth of the finite is ideality (Hegel saw this as the main position of his philosophy). Therefore, the finite is only a reflection of the infinite idea, a passing moment in the absolute. The infinite, the absolute (God) prevails over the finite. The Hegelian construction of the “true” B. cannot be recognized as correct. The goal of constructing the “true” B. is to obtain a present, given, this-worldly B. and thereby refute the opinion of the incomprehensibility of B. But this goal was realized at the cost of abandoning the infinite quantitative (or qualitative) progress, considered by Hegel as “bad” B. Hegel did not spare colors to depict all the wretchedness of such an understanding of B.: “boring, superficial change,” “meaningless repetition,” “boring alternation,” “repetitive sameness,” etc. Such a B. cannot be an object of philosophy. analysis, it is incomprehensible, it is not available, it does not go beyond the bounds of what should be and remains in the sphere of the finite. For Hegel, “true” philosophy—the category lying in the “foundation of philosophy”—has a different character. First of all, “true” B. is a denial of endless progress. It is concrete and entirely present. This concreteness and presence is already given in the definition, because The “truly” infinite consists in the fact that “... it remains with itself in its other, or, expressing the same thing as a process, consists in the fact that it comes to itself in its other” (Hegel, Op., t. 1, M.–L., 1929, p. 161). Self-closure, integrity, completeness, self-sufficient character - these are the basics. features of the “true” B. The image of the “bad” B. is a straight line, unlimitedly continuing in both directions, “... true infinity, bending back into itself, has in its image a line that has reached itself, which is closed and is entirely present..." (Hegel, Soch., vol. 5, M., 1937, pp. 151–52). Rejecting as metaphysical. the concept of “infinite progress”, “B. series”, Hegel did not see that it is to these forms that the real B. is ultimately reduced (but is not exhausted by them). Engels emphasized that endless progress is not just repetition, but is development, movement forward or backward, i.e. " required form movement" ("Dialectics of Nature", 1955, p. 189). In all the examples that Hegel and his followers illustrated the concept of "true" B., there is an implicit "bad" B. And this is understandable, because in otherwise, “true” B. would lose its infinite character. Any law of nature was considered by Hegel as a form of expression of “true” (qualitative) B., for the law is a form of universality, internal completeness, i.e., the form of B. (see. and Engels in “Dialectics of Nature”, 1955, pp. 185–86). processes proceeding according to this law. The rational content of Hegel's doctrine of "true" biology consists in emphasizing the fact that infinite progress, an infinite series can (though not always) be described, summarized or given by a completely finite expression ". true" B. is not at all infinite. By introducing the "true" B. Hegel tried to overcome the contradictory nature of the infinite and get rid of the difficulties associated with the analysis of B. Such an attempt could not be crowned with success. The real life of the material world cannot be correctly understood and reflected in concepts if one abandons the endless process of incessant changes and becomes so vigilant. "true" B. in its Hegelian interpretation. The concept of B. in dialectical. materialism differs from “bad” B. in that it does not imply a simple (without boundaries) repetition of the same thing, but, on the contrary, an unlimited variety of objects, forms, connections, characterized as infinite. It differs from Hegel’s “true” B. in that it does not deny the reality of endless processes, does not try to get rid of its internal. inconsistency. Despite their denials. On the other hand, Hegel’s analysis of B. is the best that pre-Marxist and non-Marxist philosophy has given on this issue. In bourgeois philosophy of the 19th and 20th centuries. the concept of B. is increasingly losing its objective content. The problem of B. is transferred to the psychological plane, to the sphere of consciousness, understood in a subjective idealistic sense. Dialectical materialism proceeds primarily from the objective real character of materialism, since it combines certain general properties material world, and the various logically distinguished types of B. reflect the department. sides B. of the material world. B. of matter in time can be characterized as potential B., because. it is an incomplete, ongoing series of changes. On the contrary, the world world in space can be considered as an actual reality, because it is present as given. On the other hand, quantities. B., B. the number of material objects is closely related to their qualities. B., with the unlimited variety of forms and properties of matter. The most important feature of B. is its inconsistency. Engels pointed out: “Infinity is a contradiction, and it is full of contradictions. It is already a contradiction that infinity is composed of only finite quantities, and yet this is exactly so... Precisely because infinity is contradiction, it represents an endless process, endlessly unfolding in time and space. The destruction of this contradiction would be the end of infinity" (Anti-Dühring, 1957, p. 49). The contradictory nature of B., reflecting the unity and struggle of opposite principles - finite and infinite - lies at the basis of dialectical materialism. understanding B. The Infinite denies the finite, but also retains it as its basis, as the basis of its unfolding. The infinite is absolute, because it is by its nature universal and inexhaustible; but it is also relative, because conditioned by the finite. The finite is relative, because it is unstable, transitory, temporary; but it is also absolute, for everything consists of finite phenomena and processes. B. is not only the sum of its finite parts. B. is a completely new quality, not reducible to final characteristics. Any thing, any process is finite because they are qualitatively separated from other things and processes and because they are localized in time and space, but they are also infinite because their properties are infinite and because their connections with the outside world are infinite. The concept of B. is closely related to other categories of materialism. dialectics, which is a consequence of the unity of the material world. Lit.: Engels F., Dialectics of Nature, M., 1955; his, Anti-Dühring, M., 1957; Melyukhin S. T., The problem of finite and infinite, M., 1958; Karmin A. S., On the dialectical-materialistic understanding of infinity, "Vestn. Leningrad. Univ." Ser. economics, philosophy and rights, vol. 1, 1959, No. 5; Svidersky V.I., On the dialectical-materialistic understanding of the inconsistency of infinity, ibid., 1956, No. 5, no. 1; him, Space and Time, M., 1958; Naan G.I., On the current state of cosmological science, in: Questions of cosmogony, vol. 6, M., 1958; Lemaitre G., L'hypoth?se de l'atome primitif, Neuch?tel, 1946; Whittaker E. T., The beginning and the end of the world, L., ; Schatzman E., Sur la dialectique du fini et de l'infini et la cosmologie, "Pens?e", 1954, No. 57. A. Bovin. Moscow.

“What we know is limited, but what we don’t know is infinite.”

Pierre-Simon Laplace (1749-1827), French scientist

Boundless love, immense happiness, vast space, permafrost, boundless ocean and even an endless lesson. In everyday life, we often call things and phenomena infinite, but often we don’t even think about true meaning this concept. Meanwhile, since ancient times, theologians, philosophers and other greatest minds of mankind have tried to understand its meaning. And only mathematicians have advanced the furthest in knowledge of what is called infinity.

What is infinity?

Much of what we see around us is perceived by us as infinity, but in reality it turns out to be completely finite things. This is how they sometimes explain to children how big infinity is: “If you collect one grain of sand every hundred years on a huge beach, then it will take forever to collect all the sand on the beach.” But in fact, the number of grains of sand is not infinite. It is physically impossible to count them, but we can say with confidence that their number does not exceed a value equal to the ratio of the mass of the Earth to the mass of one grain of sand.

Or another example. Many people think that if you stand between two mirrors, the reflection will be repeated in both mirrors, going into the distance, becoming smaller and smaller, so that it is impossible to determine where it ends. Alas, this is not infinity. What's really going on? No mirror reflects 100% of the light falling on it. A very high-quality mirror can reflect 99% of the light, but after 70 reflections only 50% of it will remain, after 140 reflections only 25% of the light will remain, etc. until there is too little light. Additionally, most mirrors are curved, so the many reflections you see end up "around the bend."

Let's look at how mathematics treats infinity. This is very different from any concept of infinity you've encountered before and requires a little imagination.

Infinity in mathematics

In mathematics there is a distinction potential And current infinity.

When they say that a certain quantity has infinite potential, they mean that it can be increased indefinitely, that is, there is always the potential for its increase.

The concept of actual infinity means an infinite quantity that already really exists “here and now.” Let us explain this using the example of an ordinary DIRECT line.

Example 1.

Potential infinity means that there is a straight line and it can be continuously extended (for example, by applying segments to it). Please note that the emphasis here is not on the fact that the line is infinite, but on the fact that it can be continued indefinitely.

Actual infinity means that the entire infinite straight line already exists in the present time. But the trouble is that not a single living person has seen an infinite straight line and is physically unable to do it! It’s one thing to be able to endlessly extend a straight line, and quite another to actually create an endless straight line. This very subtle difference distinguishes potential infinity from actual infinity. Ugh! It takes a lot of imagination to deal with these infinities! Let's look at another example.

Example 2.

Suppose you decide to construct a series of natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...

At some point you reach a very large number n and you think that this is the largest number. At this moment, your friend says that it costs him nothing to add 1 (one) to your number n and get an even larger number k = n + 1. Then you, slightly wounded, understand that nothing can stop you from adding to the number k one and get the number k+1. Is the number of such steps limited in advance? No. Of course, you and your friend may not have enough strength or time at some step m to take the next step m + 1, but potentially you or someone else can continue to build this series. In this case we get the concept of potential infinity.

If you and your friend manage to construct an infinite series of natural numbers, the elements of which are present all at once, at the same time, this will be actual infinity. But the fact is that no one can write down all the numbers - this is an indisputable fact!

Agree that potential infinity is more understandable for us, because it is easier to imagine. That's why ancient philosophers and mathematicians recognized only potential infinity, resolutely rejecting the possibility of operating with actual infinity.

Galileo's paradox

In 1638, the great Galileo asked the question: “Is infinitely many always equally infinitely many?” Or can there be greater and lesser infinities?”

He formulated a postulate that later received the name “Galileo’s Paradox”: There are as many natural numbers as there are squares of natural numbers, that is, in the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10... there are the same number of elements , how many are in the set 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

The essence of the paradox is as follows.

Some numbers are perfect squares (that is, squares of other numbers), for example: 1, 4, 9... Other numbers are not perfect squares, for example 2, 3, 5... This means that there should be more perfect squares and ordinary numbers together, than just perfect squares. Right? Right.

But on the other hand: for each number there is its exact square, and vice versa - for each exact square there is a whole square root, therefore there should be the same number of exact squares and natural numbers. Right? Right.

Galileo's reasoning came into conflict with the undeniable axiom that the whole is greater than any of its own parts. He could not answer which infinity is greater - the first or the second. Galileo believed that either he was mistaken about something, or such comparisons do not apply to infinities. In the latter he was right, since three centuries later, Georg Cantor proved that “the arithmetic of the infinite is different from the arithmetic of the finite.”

Countable infinities: part equals whole

Georg Cantor(1845-1918), the founder of set theory, began to use actual infinity in mathematics. He admitted that infinity exists all at once. And since there are infinite sets, all at once, it is possible to perform mathematical manipulations with them and even compare them. Since the words “number” and “amount” are inappropriate in the case of infinities, he introduced the term “power”. As a standard, Cantor took infinite natural numbers, which are enough to count anything, called this set countable, and its power - the power of a countable set, and began to compare it with the powers of other sets.

He proved that the set of natural numbers has as many elements as the set of even numbers! Indeed, let’s write one below the other:

1 2 3 4 5 6 7 8 9 10...

2 4 6 8 10 12 14 16 18 20...

At first glance, it seems obvious that the first set contains twice as many numbers as the second. But, on the other hand, it is clear that the second sequence is also countable, since any of its numbers ALWAYS corresponds to exactly one number in the first sequence. And vice versa! So the second sequence cannot be exhausted before the first. Therefore, these sets are equally powerful! It is similarly proven that the set of squares of natural numbers (from Galileo’s paradox) is countable and equal to the set of natural numbers. It follows that all countable infinities are of equal power.

It turns out very interesting: The set of even numbers and the set of squares of natural numbers (from Galileo’s paradox) are part of the set of natural numbers. But at the same time they are equally powerful. Therefore, PART IS EQUAL TO THE WHOLE!

Countless infinities

But not every infinity can be recalculated in the same way as we did with even numbers and squares of natural numbers. It turns out that it is impossible to count points on a segment, real numbers (expressed by all finite and infinite decimals), even all real numbers from 0 to 1. In mathematics they say that their number is uncountable.

Let's look at this using the example of a sequence of fractional numbers. Fractional numbers have a property that whole numbers do not have. There are no other integers between two consecutive integers. For example, no other integer “will fit” between 8 and 9. But if we add fractional numbers to the set of integers, this rule no longer holds. Yes, the number

will be between 8 and 9. Similarly, you can find a number located between any two numbers A and B:

Since this action can be repeated indefinitely, it can be argued that between any two real numbers there will always be an infinite number of other real numbers.

Thus, the infinity of real numbers is uncountable, and the infinity of natural numbers is countable. These infinities are not equivalent, but from an uncountable set of real numbers it is always possible to select a countable part, for example, natural or even numbers. Therefore, uncountable infinity is more powerful than countable infinity.

Infinity as a concept is the height of abstraction. In this respect, it can only be rivaled by the speed of light or a black hole. To tame the idea of ​​infinity, mathematicians have spent centuries inventing signs, images and stories that reconcile our minds with the unimaginable.

1. Infinity sign

Infinity has its own symbol: ∞. This sign is sometimes called a lemniscate. It was invented in 1655 by the Protestant pastor and mathematician John Wallis. The word "lemniscate" comes from the Latin lemniscus, which means "ribbon".

Perhaps when he came up with the infinity sign, Wallis took as a basis the symbol for the number 1000 written in Roman numerals (CIƆ or CƆ), which the Romans often used to denote the innumerability of objects. According to another version, the infinity symbol refers to omega (Ω or ω) - the last letter of the Greek alphabet.

The concept of infinity was proposed long before Wallis came up with a symbol for it. For example, the ancient Greek philosopher Anaximander introduced the concept of “apeiron,” which meant a certain boundless primal substance.

2. Aporia of Zeno

One of the most famous aporias ancient Greek philosopher Zeno is called “Achilles and the Tortoise”: the tortoise invites Achilles to run a race, on the condition that she starts moving a little earlier.

The tortoise is confident of his victory, because the moment Achilles reaches the tortoise's starting point, it will already crawl a little further, again increasing the distance between them.

Thus, although the distance will shorten, Achilles will never catch up with the tortoise. This paradox can be explained differently. Imagine that you are crossing a room, covering half the remaining distance with each step. First your step will be half the total distance, then a quarter, then 1/8th, 1/16th, etc. Although with each next step you will be closer to the opposite wall of the room, it is impossible to reach the end: you will need to take an infinite number of steps.

3. Pi

Another example of infinity is the number π: mathematicians use a special symbol for it because it consists of an infinite number of digits. Most often it is shortened to 3.14 or 3.14159, but no matter how many decimal places there are, it is completely impossible to write this number down.

4. The Infinite Monkey Theorem

This theorem states that if an abstract monkey punches the keys of a typewriter indefinitely, sooner or later it will type Shakespeare's Hamlet. Although some see this theorem as proof that anything is possible, mathematicians typically use it as an example of an event with a very low probability.

5. Fractals

A fractal is an abstract mathematical object, also used to depict phenomena of natural origin. In mathematics, this is a set that has the property of self-similarity: its parts are similar to the whole. Visually, such an object is a figure where the same motif is repeated on a successively decreasing scale. Therefore, the fractal image can be infinitely zoomed in: as the scale increases, more and more details appear.

When written as a mathematical equation, most fractals are non-differentiable functions.

6. Dimensions of infinity

Although infinity has no boundaries, it can have different sizes. Positive and negative numbers represent two infinite sets of equal size. However, what happens if you add these two sets? The result will be something twice the size of each of them.

In a similar way one can consider even numbers: This is also an infinite set, however it is half the size of the set of all positive numbers.

In addition, you can try adding one to infinity and make sure that the number ∞ + 1 will always be greater than ∞.

7. Cosmology and infinity

Cosmologists continue to study the Universe and ponder the concept of infinity. Is space infinite? There is still no answer to this question. Even if our physical Universe is finite, there is a possibility that it is only one Universe among many!

8. Division by zero

We know from school that division by zero is an arithmetically forbidden trick. The number 1 divided by 0 cannot be determined: any calculator will give an error code. However, according to another theory, 1/0 is a completely valid form of infinity.