Consider an infinite decimal...

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Let’s consider an infinite
decimal…

S. G. Gindikin

4,92617231248,9
7,16214301695743,0
1,7

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No one will embrace the immensity.
Others actually claim that everyone’s life is written in the book of Genesis.

Kozma Prutkov

“Real numbers are called infinite decimal fractions.”* Having given this definition, we begin from time to time to pronounce the phrase in the title of the stat. What does it mean? Are we not contradicting the worldly wisdom of Kozma Prutkov, or a similar statement by the great French mathematician A. Poincaré, which appears as epigraphs to the statue, when we intend to consider the infinite decimal fraction? The final decimal fraction can be “examined” by, for example, writing it on the board and looking at what is written. In the case of an infinite fraction, we are, of course, deprived of such an opportunity. This is due, to varying degrees, to the limitations of the size of the chalkboard, man of life, our Galaxy, etc. However, let us agree that in this statue we will focus on the mythical giant, which does not care about all these limitations and any finite considerations are within our power.

On the other hand, it is not difficult to understand that the infinity of the decimal fraction is not yet necessarily acquainted with “immenseness.” Thus, when studying rational numbers, their representation in the form of infinite periodic decimal fractions is sometimes used, and to specify the latter it is enough to indicate a finite number of decimal places preceding the period, and the period itself (the same finite number of digits). As a result, we can consider such fractions without conflicting with our “finitude.” Arthmetic and geometric progressions (infinite sequences!) are determined by two numbers (the first term and the difference or denominator). You probably know other examples of finite sequences, the definition of which requires only finite information (for example, the formula of a common term). All this suggests that the criminal, from the point of view of Kozma Prutkov, of the phrase given in the title, is not completely unconditional, and therefore requires a more thorough study, to which we move on.

So, we know that there are finite decimal fractions that can be described finitely. Let’s agree to call an infinite decimal fraction “encompassing” if there is a “verbal portrait” of it - a finite description in Russian that is uniquely defined (i.e., only one fraction satisfies this description). If there is no “verbal portrait” of a fraction, then the fraction is called “immense”. Our goal is to prove the existence of “immense” fractions.

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Let us limit ourselves only to the additions concluded between 0 and 1, that is, expressions of the form

$$0, \alpha_1 \alpha_2 \dots \alpha_k \dots,$$

(1) where $\alpha_k$, $1 \leq k < \infty$, are some numbers 0, 1, …, 8, 9. In the theory of real numbers, there is quite a lot of trouble with the fact that any additional number with a “tail” of nines (all signs, starting from a certain place, are equal to 9) is equal to some additional number with a “tail” of zeros (for example, 0.099… = 0.100…). To avoid them, we will not consider dobies that contain nine in their entries. So, we will consider dobes of the form (1), that is, sequences of the number $0, \alpha_1 \alpha_2 \dots \alpha_k \dots$, where $\alpha_k$ can be equal to 0, 1, 2, 3, 4, 5, 6, 7, 8*). We must know that even among such great ones there are “immense ones.”

How can you prove this?

Let us first consider a similar situation, relative, however, to finite sets (for better understanding we describe it in a humorous form). The director of the cinema learned that there were spectators at the screening, and the seller had ninety tickets at the box office. You probably won’t be surprised that from a comparison of these two facts he concluded that there were free riders in the hall. He simply noticed that there were more spectators than tickets, and there could not be enough tickets for all the spectators. Please note that this reasoning only makes it possible to prove the existence of a ticketless spectator, and removing a specific “hare” requires additional research (for example, checking tickets at the exit).

In all the years that have passed centuries, G. Cantor **) came up with a wonderful way to carry out similar reasoning for the free-rider age G. Cantor **) came up with a wonderful way to carry out similar reasoning for the free-ride age G. Cantor (1845-1918) - the creator of the theory of free-rider sets.

*) The perceptive reader will certainly notice when reading the further text that all considerations would only be meaningful if we limited them to only two numbers, for example, 0 and 1.

**) G. Cantor (1845-1918) - creator of the theory of free-riding sets.

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Example.
No. Numbered fractions β = 0,

1. 0, $\frac{1}{2}$ 2 3 4 5 6 7 8 0 1…
2. 0, $\frac{1}{3}$ 6 1 4 5 3 2 1 4…
3. 0, 1 $\frac{2}{3}$ 4 5 3 1 7 8 2…
4. 0, 0 1 $\frac{4}{3}$ 7 2 7 8 0 1…
5. 0, 4 2 3 $\frac{1}{2}$ 1 4 5 6 0…
6. 0, 5 6 7 2 $\frac{4}{5}$ 8 0 1 2…
7. 0, 2 4 3 5 6 7 $\frac{8}{10}$ 0 2 4…
8. 0, 7 5 4 6 2 1 $\frac{2}{10}$ 0 1 2…
9. 0, 8 0 8 0 1 0 2 $\frac{2}{20}$ 2…

β = 0,253736013…

Problem 1. Prove that the constructed fraction β is not included in the original sequence.

So, the second fact has been proven, let’s move on to the first.

Let us first discuss the venerable story of the universal library.

How many volumes should a library contain, including all kinds of books that have ever been written, are now being written, or will be written in the future? As a rule, a person who hears this question for the first time is surprised that such a universal library consists of a finite number of books (cf. the second aphorism of K. Prutkov). Each of these contradicts our idea of ​​limitless human progress… But let’s turn to the calculations. We will assume that all books consist of 500 pages (otherwise we will divide them into volumes) and that each page consists of 40 lines of 50 characters each. Now let’s discuss the issue of the printing base. We will be interested in books in Russian. For typing, you will need both characters (letters) corresponding to letters, punctuation marks, and some characters for typing special texts (however, if you do not save space, you can replace formulas, tables, etc. with their verbal descriptions). One way or another, we will assume that the capital letter is sufficient. It is convenient for us to introduce a special space character between words. By typing this character many times in a row, you can create arbitrarily long spaces (in part, thanks to this, among the five-page books you can include books consisting of a smaller number of non-blank pages). As a result, a book can be imagined as a sequence of 50 × 40 × 500 = 10⁶ characters, each of which is one of 100 characters of the typesetting alphabet (letters), i.e., like one word out of a million letters in a language whose alphabet consists of 100 letters. Note that this reduction is made possible by the detriment of the space character, otherwise the sequence of characters might not uniquely identify the book (it can be broken down into words in different ways).

Problem 2. Prove that the number of different words of length n in a language whose alphabet consists of k letters is equal to kⁿ*).

Advice. First, consider an example of our decimal number system: there are 10 different numbers of n digits (from 000…0 to 999…9).

From problem 2 it follows that the number of volumes of the universal library is written “only” as one with two million zeros. These volumes will include all the heights of wisdom, including those not comprehended by mankind, but most of the books will contain meaningless texts. There will be a book consisting of only the letters “a” (however, a book of one dot, a book and an “empty” book, entirely states of space characters).

Now let’s agree on forming a “verbal portrait” of the decimal fraction in the form of a book. Will these books be contained in the universal library? Generally speaking, no. The fact is that, speaking about the universal library, we all proceeded from the assumption that books have dimensions that are realistic from the point of view…

*) A word is simply a finite sequence of letters. We are not interested in the question of its meaningfulness.

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of our everyday practice.
However, if we refuse realization in this final refuge (and remember, we are focusing on a giant that is not inherent in it), we cannot exclude the possibility that some “verbal portraits” may occupy books that are larger in size than books from the universal library. In this regard, we introduce the concept of an ultra-pervasive library. This is what we will call a library containing all the books in as many sizes as possible. The size of a book is the number of characters in it. Let’s agree to arrange books according to the following principle: on the shelf with number $N$ there will be all books of size $N$. In particular, the entire universal library will modestly occupy one shelf with the number $N=10^6$. So, the ultra-pervasive library will already consist of an infinite number of volumes: it will have an infinite number of shelves. However, there will be a finite number of books on each shelf. From problem 2 it is clear that on the shelf numbered $N$ there are $100^N$ books. And from this it follows that all books can be rewired!

Let’s carry out the inventory as follows. We first fix the order of books on the shelves of the ultra-pervasive library. To do this, we first indicate the order of the letters in our alphabet (there are a finite number of them - 100), and then we arrange the books in the so-called lexicographic order adopted in dictionaries (of two kigg words, the one with the number of the first letter is lower; if the first letters coincide, then the second letters of the word are compared, etc.). Having arranged the books on the shelves, let’s start counting them from the first shelf (on it there are books consisting of one letter!). Having finished reducing the next shelf, we move on to the next one. Since there are books on every shelf, of course, every book will sooner or later receive a number. Inventory completed! Now let us note that the ultra-pervasive library contains verbal portraits of all “comprehensive” decimal fractions, and therefore, moving through the numbering of books in the library, we can read all these portraits, and therefore the “comprehensive” fractions (we must remember that one fraction can have several “verbal portraits”, and see if this fraction has already been assigned a number earlier; however, not It’s scary if we use the same fraction several times).

The existence of “immense” fractions has been proven.

For those who have read N. Ya. Vilenkin’s book “Stories about Sets,” we interpret the result obtained somewhat differently. Remember how the extraordinary telephone discovered by Shchon Tikhim transmitted telephone messages consisting of an infinite sequence of dots and the like? Imagine that on another telephone, we receive these telephones, they wanted to come up with a way to decipher them so that each endless telephone corresponds to a finite word in some alphabet. We have proven that such a method does not exist (words can always be confused, but telephones cannot). We can also say that there is no way to give all points of a single segment different names (points).

We have proven that “immense” fractions exist, but, like the director of the cinema, we are not able to present a specific “immense” fraction. What if you try? Let’s use the diagonal process that once helped us out. After all, we introduced the order of books in the library, which means that the number of “extensive” fractions has become fixed. Using fact 2, we will construct an uninfinite infinite decimal fraction $\beta$.

So, we went to the director of the book theater, presenting a specific “immense” fraction. But what happened? For the “immense” fraction, by definition, there should not be a finite description, but for the “immense” fraction $\beta$ such a description has been constructed; it is the previous text of this article (after all, you have no doubt that it is finite?). Only the spelled-out alphabet is not presented, but there is no doubt about that either.

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telling the truth, otherwise you may get bogged down in the difficulties associated with accurately defining the concept of “liar.” The above statement cannot be true, since then the Critian Epimenides could not tell the truth. So, it is a lie, which means it proves the existence of at least one truthful Cretan. What happened? Due to the fact that one Kritian (the fact that he was a sage is not important now) uttered a thought that, moreover, could not be true, a conclusion is drawn about the existence of a truthful Kritian. Can we really be sure of its existence? Is it worth saying after this that we are not talking about the possibility of identifying a specific truthful critic?

It would be even worse if Epimenides (like other statements) directly stated: “I am a liar” or “I am telling a lie now.” These statements can be neither true nor false. And the source of the confusion here is that in order to resolve the question of the truth of a statement, one must already know the answer to this question (a vicious circle!).

In order to eliminate the possibility of a vicious circle in “verbal portraits,” we will impose an additional requirement of efficiency, that is, we will require that the “portrait” provide the “opportunity” to explicitly construct a fraction. More precisely, we will assume that it must contain a method (or, as they say, an algorithm) that allows our machine, with unlimited memory, from which it can use arbitrarily large but finite parts, to find successive fraction signs in finite periods of time. Great is the idea of ​​a mathematical machine with unlimited memory, and the description of a fraction is a finite program for calculating its successive signs. We, of course, do not have to strictly define this great machine, and it is important that the reader understands this. Then I ask him to believe that such a definition can be given (putting that aside for another article), and, using his intuitions, to engage in a discussion of some general principles.

Now we will understand by an “encompassing” fraction a fraction for which there is a program for calculating the signs. Let’s see what has changed. Since programs can be published in book form, they are contained in an ultra-universal library, and therefore

1. “immense” fractions exist.

Further, although we have learned through bitter experience, we are wrong to refuse attempts to teach a bike to write out a specific fraction. What prevents this? We would be able to calculate the number of a book in the transuniversal library (and also write a book using the number) if we have sufficient memory and time, which means that teaching this is great. Teger will only review books containing programs for calculating fractions. To find the kth sign of the “immense” fraction β, we take the kth program, enter it into the machine, wait until it calculates the kth sign, and then connect the program for replacing signs according to the table (this is already quite easy). So, is there a contradiction again? Look, have we really built the program? You, of course, have discovered a weak point: you need to use the book to determine whether it is a program for calculating the signs of an infinite decimal fraction. And who should do this? The machine needs to be taught this. Let’s figure it out. If there was a way or program (algorithm) to find out whether a given program text (algorithm) is for calculating successive digits of an infinite decimal fraction.

Think about this result! Assuming that there is a correct definition of the concept of an algorithm, without any additional information about this definition, we