Mathematics is very simple. Mathematicians make mathematics difficult.
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If you take a closer look at all the manipulations, the error will immediately become visible. The example is written on one line and "equality" is based on the optical illusion of similarity between two different mathematical expressions. Let's write down all actions in a column, starting with the equality 1-S = S.

False equality

As you can see from this example, on opposite sides of the equal sign there are a different number of terms, on the left there is one more term. Equality cannot be fulfilled if this term is not equal to zero. So the given "proof" is a common lie. I have no doubt that other "proofs" of the sum of this infinite series are also based on false statements.

Concluding the discussion of sets, you need to consider an infinite set. The fact is that the concept of "infinity" acts on mathematicians, like a boa constrictor on a rabbit. A trembling horror of infinity deprives mathematicians of common sense. Here is an example:

Infinite set

Alpha denotes a real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take an infinite number of natural numbers as an example, then the considered examples can be represented in the following form:

The set of natural numbers

To illustrate their point, mathematicians have come up with many different methods. Personally, I look at all these methods, as the dances of shamans with tambourines. In fact, they all boil down to the fact that either part of the rooms are not occupied and new guests are accommodated in them, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I outlined my view on such decisions here in the form of a fantastic story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors requires an infinitely long time. After we vacated the first room for the guest, one of the visitors will always walk along the corridor from his room to the next one until the end of the century. Of course, the time factor can be stupidly ignored. It is mathematicians who are trying to manipulate hotel room numbers, convincing us that you can insert something that, in principle, cannot be inserted.

I will demonstrate the logic of my reasoning to you as an example of an infinite number of natural numbers. First you need to answer a very simple question: how many sets of natural numbers exist - one or many? The correct answer to this question does not exist, since we thought up the numbers ourselves; in Nature there are no numbers. Yes, Nature knows how to count, but for this she uses other mathematical tools that are not familiar to us. As Nature believes, I will tell you another time. Since we came up with numbers, we will decide how many sets of natural numbers exist. Consider both options, as befits a real scientist.

Option one. "Let us be given" one single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. Everything, there are no other natural numbers on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have one. And if you really want to? No problem. We can take a unit from the set already taken by us and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite number of natural numbers. All our manipulations can be written like this:

One set of natural numbers

I wrote down the actions in the algebraic notation and in the notation adopted in set theory, with a detailed listing of the elements of the set. The subscript indicates that the set of positive integers is one and only. It turns out that the set of natural numbers will remain unchanged only if you subtract unit from it and add the same unit.

Option two. We have on the shelf many different infinite sets of natural numbers. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take unit from another set of natural numbers and add to the set we have already taken. We can even add up two sets of natural numbers. Here's what we get:

Many sets of natural numbers

The subscripts “one” and “two” indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If we add another infinite set to one infinite set, the result is a new infinite set consisting of elements of the first two sets.

Set of natural numbers are used for counting in the same way as a ruler for measurements. Now imagine that you added one centimeter to a ruler. This will be a different a ruler, not equal to the original.

You can accept or not accept my reasoning - this is your personal matter. But if you ever run into mathematical problems, think about whether you are following the path of false reasoning, trodden by generations of mathematicians. After all, classes in mathematics, first of all, form a stable stereotype of thinking in us, and only then they add mental abilities to us (or vice versa, deprive us of free thinking).

How to divide a set into subsets? To do this, it is necessary to introduce a new unit of measure that is present in some elements of the selected set. Consider an example.

Suppose we have a set of A, consisting of four people. This set is formed on the basis of "people." We denote the elements of this set by the letter "a", the subscript with a number will indicate the serial number of each person in this set. We introduce a new unit of measurement "gender" and denote it with the letter "b". Since sexual characteristics are common to all people, we multiply each element of the set A by sexual characteristic b. Please note that now our multitude of “people” has turned into a multitude of “people with sexual characteristics”. After that, we can divide the sexual characteristics into male bm and female bw sexual characteristics. Now we can apply the mathematical filter: we choose one of these sexual characteristics, no matter which one - male or female. If it is present in a person, then we multiply it by one; if there is no such sign, we multiply it by zero. And then we use ordinary school math. See what happened.

Set and subset

After multiplication, contractions, and rearrangements, we got two subsets: a subset of Bm men and a subset of Bw women. Mathematicians reason the same way when they apply set theory in practice. But they don’t dedicate us in detail, but give the finished result - "a set of people consist of a subset of men and a subset of women." Naturally, you may wonder how correctly mathematics is applied in the above transformations? I dare to assure you that, in essence, everything was done correctly; it’s enough to know the mathematical justification of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time, I'll tell you about it.

You can combine two sets into one by selecting the unit of measure present in the elements of these two sets.

As you can see, units of measure and ordinary mathematics turn set theory into a relic of the past. A sign that the set theory is not all right is that for set theory mathematicians have come up with their own language and their own notation. Mathematicians did what shamans once did. Only shamans know how to "apply" their "knowledge" correctly. They teach us this "knowledge".

In conclusion, I want to show you how mathematicians manipulate infinite sets.

I have already told you that set theory is a herd theory by which shamans try to sort out "sea urchins" of reality. How do they do that? How does the formation of the set actually occur?

Initially, individual elements stand out from reality. After that, a set are formed from some elements that have a common property. Shamans always know in advance how a set they want to show us.

I will show the process by example. We select "red solid with holes" - these are our elements. At the same time, we see that these elements are with a bow, and there are without a bow. After that, we select some of the elements and form a set of "with a bow." This is how shamans get their food, tying their set theory to reality.

Now let's take a “solid with holes with a bow” and combine these elements by color, selecting red ones. We got a set of red. Now the question is: the resulting sets "with a bow" and "red" - is it the same set or two different sets? Only shamans know the answer.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid with holes with a bow". The formation took place in four different units of measurement: color (red), strength (solid), integrity (with holes), jewelry (with a bow). Only a set of units of measurement allows us to adequately describe real objects in the language of mathematics. Here is how it looks.

The formation of the set

The letter "a" with different indices indicates different units of measurement. In brackets, the units of measurement that are considered at the preliminary stage are highlighted. The unit of measure by which the set is formed is the set out of brackets. The last line shows the final result - an element of the set. If we use units of measure to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dancing of shamans with tambourines. Shamans can "intuitively" reach the same result, arguing for its "evidence", because units of measurement are not included in their "scientific" arsenal.

Using units of measure, it is very easy to split one set into several subsets, or to combine several sets into one set. Let's take a closer look at the algebra of this process.

If mathematicians cannot reduce the concept of "set" to other concepts, then they do not understand anything in mathematics. I answer the question: how are the elements of one set different from the elements of another set? The answer is very simple: numbers and units of measurements.

Today everything belongs to some kind of set (as mathematicians assure us). By the way, did you look at yourself in the mirror? Have you seen on your forehead a list of those sets to which you belong? And I have not seen such a list. I will say more - not a single thing in reality has a tag with a list of sets to which this thing belongs. Sets are shamans' inventions. How do they do it? Let's recall the history and see how the elements of the set looked like before the mathematician-shamans pulled them into their sets.

Once upon a time, when no one had heard of mathematics, and only trees and Saturn had rings, huge herds of wild elements of sets roamed the physical fields (after all, shamans had not yet invented mathematical fields). They looked something like this.

Sea urhin

Yes, do not be surprised, from the point of view of mathematics, all the elements of sets are most similar to sea urchins - from one point, like needles, units of measurements stick out in all directions. For those who missed something, I remind you that any unit of measure can be represented geometrically as a segment of arbitrary length, and a number as a point. Geometrically, any value can be represented as a bunch of segments sticking out in different directions from one point. This point is point zero. I will not draw this work of geometric art (no inspiration), but you can easily imagine it.

What units of measurements form an element of the set? Any, describing this element from different points of view. These are the ancient units of measurement used by our ancestors and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are units of measurement unknown to us that our descendants will come up with and which they will use to describe reality.

We figured out the geometry - the proposed model of the elements of the set has a clear geometric representation. What about physics? Units of measurement - this is the direct connection of mathematics with physics. If shamans do not recognize units of measurement as a full-fledged element of mathematical theories, this is their problem. Personally, I can’t imagine real math science without units of measurement. That is why, at the very beginning of the story about set theory, I spoke of it as a Stone Age.

But let us pass to the most interesting thing - to the algebra of elements of sets. Algebraically, any element of a set is a collection of different quantities. It looks like this.

Element of the set

I deliberately do not apply the conventions adopted in set theory, since we consider the element of the set in its natural environment before the advent of set theory. Each pair of letters designates a separate quantity consisting of a number indicated by the letter "n" and a unit of measurement indicated by the letter "a". Indexes indicate that the numbers and units are different. One element of the set can consist of an infinite number of quantities (how much imagination we have and our descendants). Each value is geometrically depicted as a separate segment. In the example with the sea urchin, one value is one needle.

How do shamans form sets of different elements? In fact, by units of measurement or by numbers. Without understanding anything in mathematics, they take different sea urchins and carefully examine them in search of the only needle by which they form a multitude. If there is such a needle, then this element belongs to the set, if there is no such needle, it is an element of another set. Shamans tell us tales about thought processes and a single whole.

As you may have guessed, one and the same element can belong to very different sets. Further I will show you how the sets, subsets and other attributes of shamans are formed.

The concepts of “set” and “multiset” are two trump ace in the sleeves of a sharpie, one in each sleeve. The sharpie takes out from the sleeve that trump ace, which is more convenient for him to get. In any case, the sharpie always wins. In the same way, any mathematician will always justify set theory.

There can’t be two identical elements in a set, but if there are identical elements in a set, such a set is called a “multiset.” Intelligent beings can never understand such a logic of absurdity. This is the level of talking parrots and trained monkeys. Mathematicians act as ordinary animal trainer, preaching to us their absurd ideas.

Once the engineers who built the bridge, during the testing of the bridge, were in a boat under the bridge. If the bridge collapsed, a mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer built other bridges.

No matter how mathematicians hide behind the phrase “mathematics studies abstract concepts”, there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. We apply the mathematical set theory to the mathematicians themselves.

We studied mathematics very well and now we are giving out salaries. Here comes a mathematician for his money. We count the whole amount to him and lay out on his table on different piles, into which we place banknotes of the same denomination. Then we take one banknote from each pile and hand over to the mathematician his "mathematical set of salary." We explain the mathematics that he will receive the remaining bills only when he proves that a set without the same elements is not equal to a set with the same elements. After that, the most interesting will happen.

Set and multiset

First of all, the logic of the deputies will work: "It can be applied to others, but not to me!" Further, the mathematician will assure us that on banknotes of the same denomination there are different numbers of banknotes, therefore they cannot be considered the same elements of the set. Well, we count the salary in coins - there are no numbers on the coins. Here, the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms of each coin is unique ...

And now I have the most interesting question: where is the line behind which the elements of the multiset turn into the elements of the set and vice versa? Such a facet does not exist - shamans decide everything, science is absent here.

You see that the same set of elements is both a multitude and a multiset at the same time. How right? And here the mathematician-shaman-schuller takes an ace from his sleeve and begins to tell us either about the multitude or the multiset. In any case, he will convince us that the theory of sets is correct.

To understand how modern shamans apply set theory to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you how to do it.

Set theory is the stone age of mathematics. Only shamans know how to properly form the set. Let's look at set theory from the side and reveal some of the secrets of shamans. To understand the essence of set theory, it is necessary to consider another mathematical concept - a function. The function sets the correspondence between the elements of two sets. Each element from the herd of hunters is associated with one or more elements from the herd of prey, each element from the herd of prey is assigned one or more elements from the herd of hunters. Only shamans know what kind of herd belongs to and how to divide prey properly.

What is set theory? This is a herd theory. What will happen if we combine a herd (set) mathematicians and a herd (set) sheep: sheeps with a mathematical education or mathematicians with sheep’s brains? I don’t know what set theory says about the results of such a union, but in reality mathematicians will have a great barbecue picnic. I have nothing against set theory, as one of the mathematical tools. But to stand up for “scientific” theories based on such a primitive and vague concept is already too much.

Any theory should be tested by practice, even mathematical. I will show you an example of the practical application of set theory by the example of such concepts as “set” and “multiset”.