The concept of a set is the original not strictly defined concept. We give here the definition of a set (more precisely, an explanation of the idea of ​​a set) belonging to G. Cantor: “Under a variety or a set, I mean in general all the many things that can be thought of as a single one, i.e. such a set of certain elements that can be connected by means of one law into one whole."

Sets will, as a rule, be denoted by capital letters of the Latin alphabet, and their elements by small letters, although sometimes this convention will have to be deviated from, since the elements of a certain set may be other sets. The fact that an element a belongs to the set A is written as a\in A .

In mathematics, we deal with a wide variety of sets. For the elements of these sets, we use two main types of notation: constants and variables.

An individual constant (or just a constant) with range A denotes a fixed element of the set A . Such, for example, are the designations (records in a certain number system) of real numbers: 0;\,2;\,7,\!34. For two constants b and b with range A, we will write a=b , meaning by this the coincidence of the elements of the set A denoted by them.

An individual variable (or just a variable) with range A denotes an arbitrary, not predetermined element of the set A . Here we say that the variable x runs through the set A or the variable x takes arbitrary values ​​on the set A . You can fix the value of a variable x by writing x=a , where a is a constant with the same range as x . In this case, we say that instead of the variable x, its specific value a was substituted, or a was substituted for x, or the variable x took on the value a.

The equality of variables x=y is understood as follows: whenever the variable x takes on an arbitrary value a , the variable y takes on the same value a , and vice versa. Thus, equal variables "synchronously" always take on the same values.

Usually constants and variables whose range is some numerical set, namely one of the sets \mathbb(N),\, \mathbb(Z),\, \mathbb(Q),\, \mathbb(R) and \mathbb(C) are called, respectively, natural, integer (or integer), rational, real, and complex constants and variables. In the course of discrete mathematics, we will use various constants and variables, the range of which is not always a numerical set.

To shorten the record, we will use logical symbolism, which allows us to write statements briefly, like formulas. The concept of an utterance is not defined. It is only indicated that any statement can be true or false (of course, not both at the same time!).

Logical operations (bindings) on sets

To form new statements from existing statements, the following logical operations (or logical connectives) are used.

1. Disjunction \lor : the proposition P\lor Q (read: "P or Q") is true if and only if at least one of the propositions P and Q is true.

2. \land conjunction: P\land Q (read: "P and Q") is true if and only if both P and Q are true.

3. \lnot negation: \lnot P (read: "not P") is true if and only if P is false.

4. Implication \Rightarrow : the proposition P \Rightarrow Q (read: "if P then Q" or "P implies Q") is true if and only if the proposition is true or both propositions are false.

5. Equivalence (or equivalence) \Leftrightarrow : a proposition (read: "P if and only if Q") is true if and only if both propositions P and Q are either both true or both false. Any two statements P and Q such that is true P \Leftrightarrow Q, are called logically equivalent or equivalent.

Writing down sentences with logical operations, we assume that the order of execution of all operations is determined by the arrangement of brackets. To simplify notation, parentheses are often omitted, while accepting a certain order of operations ("priority convention").

The negation operation is always performed first, and therefore it is not enclosed in parentheses. The second one performs the operation of conjunction, then disjunction, and finally implication and equivalence. For example, the statement (\lnot P)\lor Q is written as \lnot P\lor Q . This proposition is the disjunction of two propositions: the first is the negation of P and the second is the negation of Q. In contrast, the proposition \lnot (P\lor Q) is the negation of the disjunction of the propositions P and Q .

For example, the statement \lnot P\land Q\lor\lnot Q\land P \Rightarrow\lnot Q after placing the brackets in accordance with the priorities, it will take the form

\bigl(((\lnot P)\land Q)\lor ((\lnot Q)\land P)\bigr)\Rightarrow (\lnot Q).

Let us make some comments about the logical connectives introduced above. The meaningful interpretation of disjunction, conjunction and negation does not need special explanations. The implication P \Rightarrow Q is true, by definition, whenever Q is true (regardless of P being true) or P and Q are both false. Thus, if the implication P\Rightarrow Q is true, then when P is true, Q is true, but the converse may not be true, i.e. when P is false, Q can be either true or false. This motivates the reading of the implication in the form "if P , then Q ". It is also easy to understand that the statement P\Rightarrow Q is equivalent to the statement \lnot P\lor Q and thus, meaningfully "if P , then Q " is identified with "not P or Q ".

The equivalence \Leftrightarrow is nothing but a "two-sided implication", i.e. P\Leftrightarrow Q is tantamount to (P \Rightarrow Q)\land (Q \Rightarrow P). This means that the truth of P implies the truth of Q, and vice versa, the truth of Q implies the truth of P.

Example 1.1. To determine the truth or falsity of a complex statement, depending on the truth or falsity of the statements included in it, truth tables are used.

The first two columns of the table record all possible sets of values ​​that the statements P and Q can take. The truth of the statement is indicated by the letter "I" or the number 1, and the falsity - by the letter "L" or the number 0. The remaining columns are filled in from left to right. So for each set of P and Q values, the corresponding propositional values ​​are found.

The truth tables of logical operations have the simplest form (Tables 1.1-1.5).

Consider a compound statement (\lnot P\land Q)\Rightarrow (\lnot Q\land P). For computational convenience, we denote the statement \lnot P\land Q by A , the statement \lnot Q\land P by B , and write the original statement as A \Rightarrow B . The truth table of this statement consists of columns P,\,Q,\,A,\,B and A \Rightarrow B (Table 1.6).

Predicates and quantifiers

Compound statements are formed not only through logical connectives, but also with the help of predicates and quantifiers.

A predicate is a statement containing one or more individual variables. For example, "x is even number" or "x is a student of Moscow State Technical University. Bauman, received in 1999". In the first predicate x is an integer variable, in the second - a variable running through the set of "human individuals". An example of a predicate containing several individual variables is: "x is the son of y", "x, y and z study in the same group", "x is divisible by y", "x is less than y", etc. We will write the predicates in the form P(x),\, Q(x,y),\, R(x,y,z), assuming that all the variables included in the given predicate are listed in brackets.

Substituting instead of each variable included in the predicate P(x_1,\ldots,x_n), specific value, i.e. fixing the values ​​, where a_1,\ldots,a_n are some constants with the corresponding range of values, we obtain a statement that does not contain variables. For example, "2 is an even number", "Isaac Newton is a student of the Moscow State Technical University named after Bauman, who entered in 1999", "Ivanov is the son of Petrov", "5 is divisible by 7", etc. Depending on whether the statement thus obtained is true or false, the predicate P is said to be satisfied or not satisfied on the set of values ​​of the variables x_1=a_1,\ldots,x_n=a_n. A predicate that is satisfied on any set of variables included in it is called identically true, and a predicate that is not satisfied on any set of values ​​of its variables is called identically false.

A statement from a predicate can be obtained not only by substituting the values ​​of its variables, but also by means of quantifiers. Two quantifiers are introduced - existence and universality, denoted \exists and \forall respectively.

statement (\forall x\in A)P(x)("for every element x that belongs to the set A , P(x) is true ", or, more briefly, "for all x\in A, P(x) is true ") is true, by definition, if and only if the predicate P (x) is executed for each value of the variable x .

statement (\exists x\in A)P(x)("there exists, or there is, such an element x of the set A that P(x) is true ", also "for some x\in A P(x) is true ") is true, by definition, if and only if on some values variable x, the predicate P(x) is satisfied.

Associating predicate variables with quantifiers

When a statement is formed from a predicate by means of a quantifier, it is said that the variable of the predicate is bound by the quantifier. Similarly, variables are bound in predicates containing several variables. In the general case, expressions of the form

(Q_1x_1\in A_1)(Q_2x_2\in A_2)\ldots (Q_nx_n\in A_n) P(x_1,x_2, \ldots, x_n),

where any of the quantifiers \forall or \exists can be substituted for each letter Q with index.

For example, the statement (\forall x\in A)(\exists y\in B)P(x,y) reads like this: "for every x\in A there exists y\in B such that P(x,y) is true ". If the sets that run through the predicate variables are fixed (meaning "by default"), then the quantifiers are written in the abbreviated form: (\forall x)P(x) or (\exists x)P(x) .

Note that many mathematical theorems can be written in a form similar to the quantifier statements just given, for example: "for all f and for all a true: if f is a function differentiable at a, then the function f is continuous at a".

Ways of specifying sets

Having discussed the features of the use of logical symbolism, let us return to the consideration of sets.

Two sets A and B are considered equal if any element x of set A is an element of set B and vice versa. It follows from the above definition of equal sets that a set is completely determined by its elements.

Let us consider ways of specifying concrete sets. For a finite set, the number of elements of which is relatively small, the method of direct enumeration of elements can be used. The elements of a finite set are listed in curly braces in an arbitrary fixed order\(1;3;5\) . We emphasize that since a set is completely determined by its elements, then when a finite set is specified, the order in which its elements are listed does not matter. Therefore records \{1;3;5\},\, \{3;1;5\},\, \{5;3;1\} etc. all define the same set. In addition, sometimes repetitions of elements are used in the notation of sets. We will assume that the notation \(1;3;3;5;5\) defines the same set as the notation \(1;3;5\) .

In the general case, for a finite set, the notation is used. As a rule, repetition of elements is avoided. Then the finite set given by the notation \(a_1,\ldots,a_n\), consists of n elements. It is also called an n-element set.

However, the method of specifying a set by directly enumerating its elements is applicable in a very narrow range of finite sets. The most general way to specify concrete sets is to specify some property that all elements of the described set must have, and only they.

This idea is implemented in the following way. Let the variable x run through some set U , called the universal set. We assume that only such sets are considered whose elements are also elements of the set U . In this case, a property that only the elements of a given set A have can be expressed by means of the predicate P(x) , which is executed if and only if the variable x takes an arbitrary value from the set A . In other words, P(x) is true if and only if the individual constant a\in A is substituted for x.

The predicate P is called in this case the characteristic predicate of the set A , and the property expressed using this predicate is called the characteristic property or collectivizing property.

The set defined through the characteristic predicate is written in the following form:

A=\bigl\(x\colon~ P(x)\bigr\).

For example, A=\(x\in\mathbb(N)\colon\, 2x\) means that "A is the set consisting of all elements x such that each of them is an even natural number".

The term "collectivizing property" is motivated by the fact that this property allows you to collect disparate elements into a single whole. Thus, the property that defines the set G (see below) literally forms a kind of "collective":

If we return to Cantor's definition of a set, then the characteristic predicate of a set is the law by which a set of elements is combined into a single whole. A predicate specifying a collectivizing property can be identically false. A set defined in this way will have no elements. It is called the empty set and denoted by \varnothing .

In contrast, an identically true characteristic predicate defines a universal set.

Note that not every predicate expresses some collectivizing property.

Remark 1.1. The specific content of the concept of a universal set is determined by the fact specific context, in which we apply set-theoretic ideas. For example, if we deal only with various numerical sets, then the set \mathbb(R) of all real numbers can appear as a universal one. Each branch of mathematics deals with a relatively limited set of sets. Therefore, it is convenient to assume that the elements of each of these sets are also the elements of some universal set "embracing" them. By fixing the universal set, we thereby fix the range of values ​​of all the variables and constants that appear in our mathematical reasoning. In this case, it is precisely possible not to indicate in the quantifiers the set that runs through the variable bound by the quantifier. In what follows, we will meet with various examples of concrete universal sets.

CANTOR'S SET THEORY. Kantor developed a certain technique for operating with actually infinite sets and constructed a certain analogue of the concept of quantity for infinite sets. The basis of this technique is the concept of a one-to-one correspondence between elements of two sets. They say that the elements of two sets can be put in a one-to-one correspondence if each element of the first set can be associated with an element of the second set, different - different, and at the same time, each element of the second set will correspond to some element of the first. Such sets are said to be equivalent, that they have the same cardinality, or the same cardinal number. If it can be proved that the elements of the set A can be put in a one-to-one correspondence with the elements of the subset B1 of the set B, and the elements of the set B cannot be put in a one-to-one correspondence with the elements of A, then they say that the cardinality of the set B is greater than the cardinality of the set A. These the definitions apply to finite sets as well. In this case, power is analogous to finite numbers. But infinite sets have paradoxical properties in this sense. An infinite set turns out to be equivalent to its part, for example. the way it happens in the so-called. Galileo's Paradox:

1, 2, 3, 4, ..., n, ...

2, 4, 6, 8, ..., 2n, ...

These paradoxes have been known for a long time, and it is they, in particular, that have served as an obstacle to the consideration of actually infinite sets. Bolzano explained in Paradoxes of the Infinite that the specificity of the actually infinite simply affects here. Dedekind considered this property of actually infinite sets to be characteristic.

Cantor develops the arithmetic of cardinal numbers. The sum of two cardinal numbers is the cardinality of the union of the sets corresponding to them, the product is the cardinality of the so-called. sets-products of two given sets, and so on. The most important is the transition from the given set to the set-degree, i.e., by definition, to the set of all subsets of the original set. Cantor proves a fundamental theorem for his theory: the cardinality of a set-degree is greater than the cardinality of the original set. If the power of the original set is written in terms of a, then, in accordance with the arithmetic of cardinal numbers, the power of the set-degree will be 2a, and we have, therefore, 2a >a.

So, passing from some infinite set, e.g. from all the many natural numbers with cardinality ℵα (Cantor's notation) to the set of all subsets of this set, to the set of all subsets of this new set, etc., we will get a series of sets of ever increasing cardinality. Is there any limit to this increase? This question can be answered only by introducing some additional concepts.

Generally speaking, it is impossible to operate with infinite sets devoid of any additional structure. Therefore, Cantor introduced ordered sets into consideration, i.e. sets, for any two elements of which the relation "greater than" > (or "less than"<). Это отношение должно быть транзитивным: из a < b и b < с следует: а < с. Собственно, наиболее продуктивным для теории множеств является еще более узкий класс множеств: вполне упорядоченные множества. Так называются упорядоченные множества, у которых каждое подмножество имеет наименьший элемент. Вполне упорядоченные множества легко сравнивать между собой: они отображаются одно на часть другого с сохранением порядка. Символы вполне упорядоченных множеств, или ординальные (порядковые) числа, также образуют вполне упорядоченное множество, и для них также можно определить арифметические действия: сложение (вычитание), умножение, возведение в степень. Ординальные числа играют для бесконечных множеств роль порядковых чисел, кардинальные – роль количественных. Множество (бесконечное) определенной мощности можно вполне упорядочить бесконечным числом способов, каждому из которых будет соответствовать свое ординальное число. Тем самым каждому кардиналу (Кантор ввел для обозначения кардиналов «алефы» – первую букву еврейского алфавита с индексами) ℵα будет соответствовать бесконечно много ординалов:

0 1 2 ... ω0, ω0 + 1 ... ω1... ω2 ... ωn ... ωω0 ... Ω (ordinals)

0 1 2 ... ℵ0 ... ℵ1 ... ℵ2 ℵn …ℵ ω0 … τ (“tau”-cardinals)

According to the theorems of set theory, any "segment" of the scale Ω of ordinal numbers, itself as a completely ordered set, will have a larger ordinal than all contained in this segment. This implies that it is impossible to consider all Ω as a set, because otherwise Ω would have as its ordinal β, which is greater than all ordinals in Ω, but since the latter contains all ordinals, i.e. and β, then it would be: β > β (the Burali–Forti paradox, 1897). Kantor sought to circumvent this paradox by introducing (since the 1880s) the concept of consistency. Not every plurality (Vielheit) is a plurality (Menge). A plurality is called consistent, or a plurality if it can be considered as a complete whole. If the assumption of "joint existence" of all elements of the multiplicity leads to a contradiction, then the multiplicity turns out to be inconsistent, and, in fact, it cannot be considered in set theory. Such inconsistent sets are, in particular, Ω, the set of all ordinal numbers, and τ (“tau”), the set of all cardinals (“alefs”). Thus, we again return to infinity as a process. As the mathematician of the 20th century writes, P. Vopenka: “The theory of sets, whose efforts were directed at the actualization of potential infinity, turned out to be unable to eliminate potentiality, but only managed to move it to a higher sphere” (Vopenka P. Mathematics in alternative set theory. - “New in foreign science. Mathematics ”, 1983, No. 31, p. 124.) This did not, however, embarrass Kantor himself. He believed that the scale of “alephs” rises to the infinity of God himself, and therefore the fact that the latter turns out to be mathematically inexpressible was for him self-evident: “I never proceeded from any “Genus supremum” of actual infinity. Quite the contrary, I have rigorously proved the absolute non-existence of "Genus supremum" for actual infinity. That which transcends everything infinite and transfinite is not "Genus"; it is the only, highly individual unity in which everything is included, which includes the "Absolute", incomprehensible to human understanding. This is "Actus Purissimus", which is called God by many" (Meschkowski H. Zwei unveroffentlichte Briefe Georg Cantors. - "Der Mathematilkuntemcht", 1971, No. 4, S. 30-34).

B. H. Katasonov

New Philosophical Encyclopedia. In four volumes. / Institute of Philosophy RAS. Scientific ed. advice: V.S. Stepin, A.A. Huseynov, G.Yu. Semigin. M., Thought, 2010, vol. I, A - D, p. 249-250.

I am a theoretical physicist by education, but I have a good mathematical background. In the magistracy one of the subjects was philosophy, it was necessary to choose a topic and submit a paper on it. Since most of the options were more than once obmusoleny, I decided to choose something more exotic. I do not pretend to novelty, I just managed to accumulate all / almost all available literature on this topic. Philosophers and mathematicians can throw stones at me, I will only be grateful for constructive criticism.

P.S. Very "dry language", but quite readable after the university program. For the most part, definitions of paradoxes were taken from Wikipedia (simplified wording and ready-made TeX markup).

Introduction

Both the set theory itself and the paradoxes inherent in it appeared not so long ago, just over a hundred years ago. However, during this period a long way has been traveled, the theory of sets, one way or another, actually became the basis of most sections of mathematics. Its paradoxes, connected with Cantor's infinity, were successfully explained literally in half a century.

You should start with a definition.

What is a multitude? The question is quite simple, the answer to it is quite intuitive. A set is a set of elements represented by a single object. Cantor in his work Beiträge zur Begründung der transfiniten Mengenlehre gives a definition: by “set” we mean the combination into a certain whole M of certain well-defined objects m of our contemplation or our thinking (which will be called “elements” of the set M). As you can see, the essence has not changed, the difference is only in the part that depends on the worldview of the determinant. The history of set theory, both in logic and in mathematics, is highly controversial. In fact, Kantor laid the foundation for it in the 19th century, then Russell and the others continued the work.

Paradoxes (logic and set theory) - (Greek - unexpected) - formal logical contradictions that arise in the meaningful set theory and formal logic while maintaining the logical correctness of reasoning. Paradoxes arise when two mutually exclusive (contradictory) propositions are equally provable. Paradoxes can appear both within scientific theory and in ordinary reasoning (for example, Russell's paradox about the set of all normal sets is given by Russell: "The village barber shaves all those and only those inhabitants of his village who do not shave themselves. Should he shave yourself?"). Since the formal-logical contradiction destroys reasoning as a means of discovering and proving the truth (in a theory in which a paradox appears, any sentence, both true and false, is provable), the problem arises of identifying the sources of such contradictions and finding ways to eliminate them. The problem of philosophical understanding of specific solutions to paradoxes is one of the important methodological problems formal logic and logical foundations of mathematics.

The purpose of this work is to study the paradoxes of set theory as heirs of ancient antinomies and quite logical consequences of the transition to a new level of abstraction - infinity. The task is to consider the main paradoxes, their philosophical interpretation.

Basic paradoxes of set theory

The barber only shaves people who don't shave themselves. Does he shave himself?

Let's continue with a brief excursion into history.

Some of the logical paradoxes have been known since ancient times, but due to the fact that mathematical theory was limited to arithmetic and geometry alone, it was impossible to correlate them with set theory. In the 19th century, the situation changed radically: Kantor reached a new level of abstraction in his works. He introduced the concept of infinity, thereby creating new section mathematics and thus allowing different infinities to be compared using the concept of "power of a set". However, in doing so, he created many paradoxes. The first is the so-called Burali-Forti paradox. In the mathematical literature, there are various formulations based on different terminology and an assumed set of well-known theorems. Here is one of the formal definitions.

It can be proved that if x is an arbitrary set of ordinals, then the sum-set is an ordinal greater than or equal to each of the elements x. Suppose now that is the set of all ordinal numbers. Then is an ordinal number greater than or equal to any of the numbers in . But then and is an ordinal number, moreover, it is already strictly greater, and therefore not equal to any of the numbers in . But this contradicts the condition that is the set of all ordinal numbers.

The essence of the paradox is that when the set of all ordinal numbers is formed, a new one is formed. ordinal type, which was not yet among "all" transfinite ordinal numbers that existed before the formation of the set of all ordinal numbers. This paradox was discovered by Cantor himself, independently discovered and published by the Italian mathematician Burali-Forti, the latter's errors were corrected by Russell, after which the formulation acquired its final form.

Among all attempts to avoid such paradoxes and to some extent try to explain them, the idea of ​​the already mentioned Russell deserves the most attention. He proposed to exclude from mathematics and logic impredicative sentences in which the definition of an element of a set depends on the latter, which causes paradoxes. The rule sounds like this: "no set C can contain elements m, defined only in terms of the set C, as well as elements n, assuming this set in their definition" . Such a restriction on the definition of a set allows us to avoid paradoxes, but at the same time significantly narrows the scope of its application in mathematics. In addition, this is not enough to explain their nature and reasons for their appearance, rooted in the dichotomy of thought and language, in the features of formal logic. To some extent, this restriction can be traced an analogy with what in a later period cognitive psychologists and linguists began to call "basic level categorization": the definition is reduced to the most easy-to-understand and study concept.

Assume that the set of all sets exists. In this case, it is true, that is, any set t is a subset of V. But it follows from this that the power of any set does not exceed the power of V. But by virtue of the axiom of the set of all subsets, for V, as well as for any set, there is a set of all subsets , and by Cantor's theorem, which contradicts the previous statement. Therefore, V cannot exist, which contradicts the "naive" hypothesis that any syntactically correct boolean condition defines a set, i.e. that for any formula A not containing y freely. A remarkable proof of the absence of such contradictions on the basis of the axiomatized Zermelo-Fraenkel set theory is given by Potter.

From a logical point of view, both of the above paradoxes are identical to the “Liar” or “The Barber”: the expressed judgment is directed not only to something objective in relation to him, but also to himself. However, one should pay attention not only to the logical side, but also to the concept of infinity, which is present here. The literature refers to the work of Poincaré, in which he writes: "belief in the existence of actual infinity ... makes these non-predicative definitions necessary"" .
In general, the main points are:

• in these paradoxes, the rule is violated to clearly separate the “spheres” of the predicate and the subject; the degree of confusion is close to the substitution of one concept for another;
• usually in logic it is assumed that in the process of reasoning the subject and predicate retain their scope and content, in this case
  transition from one category to another, resulting in a mismatch;
• the presence of the word "all" makes sense for a finite number of elements, but in the case of an infinite number of them, it is possible to have one that
  to define itself would require the definition of a set;
• basic logical laws are violated:
  • the law of identity is violated when the non-identity of the subject and the predicate is revealed;
  • the law of contradiction - when two contradictory judgments are derived with the same right;
  • the law of the excluded third - when this third has to be recognized, and not excluded, since neither the first nor the second can be recognized one without the other, because they are equally valid.

The third paradox bears Russell's name.. One definition is given below.
Let K be the set of all sets that do not contain themselves as their element. Does K contain itself as an element? If yes, then, by definition of K, it should not be an element of K - a contradiction. If not - then, by definition of K, it must be an element of K - again a contradiction. This statement is logically derived from Cantor's paradox, which shows their relationship. However, the philosophical essence manifests itself more clearly, since the “self-movement” of concepts takes place right “before our eyes”.

Tristram Shandy's paradox:
In Stern's The Life and Opinions of Tristram Shandy, Gentleman, the hero discovers that he needed whole year to describe the events of the first day of his life, and another year was needed to describe the second day. In this regard, the hero complains that the material of his biography will accumulate faster than he can process it, and he will never be able to complete it. “Now I maintain,” Russell objects to this, “that if he lived forever and his work would not become a burden to him, even if his life continued to be as eventful as at the beginning, then not one part of his biography would not remain unwritten.
Indeed, Shandy could describe the events of the nth day for the nth year and, thus, in his autobiography, every day would be captured.

In other words, if life lasted indefinitely, then it would have as many years as days.

Russell draws an analogy between this novel and Zeno with his tortoise. In his opinion, the solution lies in the fact that the whole is equivalent to its part at infinity. Those. leads to a contradiction only "axiom common sense» . However, the solution of the problem lies in the realm of pure mathematics. Obviously, there are two sets - years and days, between the elements of which there is a one-to-one correspondence - a bijection. Then under the condition endless life the main character has two infinite sets of equal power, which, if we consider the power as a generalization of the concept of the number of elements in a set, resolves the paradox.

Paradox (theorem) of Banach-Tarski or doubling the ball paradox- a theorem in set theory stating that a three-dimensional ball is equally composed of two of its copies.
Two subsets of the Euclidean space are called equally composed if one can be divided into a finite number of parts, moved them, and made up of them the second.
More precisely, two sets A and B are equally composed if they can be represented as a finite union of disjoint subsets such that for each i the subset is congruent.

If we use the choice theorem, then the definition sounds like this:
The axiom of choice implies that there is a division of the surface of a unit sphere into a finite number of parts, which, by transformations of the three-dimensional Euclidean space that do not change the shape of these components, can be assembled into two spheres of unit radius.

Obviously, given the requirement for these parts to be measurable, this statement is not feasible. The famous physicist Richard Feynman in his biography told how at one time he managed to win the dispute about splitting an orange into a finite number of parts and recomposing it.

At certain points this paradox is used to refute the axiom of choice, but the problem is that what we consider elementary geometry is not essential. Those concepts that we consider intuitive should be extended to the level of properties of transcendental functions.

To further weaken the confidence of those who believe that the axiom of choice is wrong, one should mention the theorem of Mazurkiewicz and Sierpinski, which states that there is a non-empty subset E of the Euclidean plane that has two disjoint subsets, each of which can be divided into a finite number of parts, so that they can be translated by isometries into a covering of the set E.
The proof does not require the use of the axiom of choice.
Further constructions based on the axiom of certainty give a resolution to the Banach-Tarski paradox, but are not of such interest.

• Richard's paradox: it is required to name " smallest number not named in this book. The contradiction is that on the one hand, this can be done, since there is the smallest number named in this book. Proceeding from it, one can also name the smallest unnamed. But here a problem arises: the continuum is uncountable, between any two numbers you can insert an infinite number of intermediate numbers. On the other hand, if we could name this number, it would automatically move from the class not mentioned in the book to the class mentioned.
• The Grelling-Nilson paradox: words or signs can denote a property and at the same time have it or not. The most trivial formulation sounds like this: is the word “heterological” (which means “not applicable to itself”) heterological?.. It is very similar to Russell’s paradox due to the presence of a dialectical contradiction: the duality of form and content is violated. In the case of words that have a high level of abstraction, it is impossible to decide whether these words are heterological.
• Skolem's paradox: using Godel's completeness theorem and the Löwenheim-Skolem theorem, we obtain that axiomatic set theory remains true even when only a countable set of sets is assumed (available) for its interpretation. In the same time
  axiomatic theory includes the already mentioned Cantor's theorem, which brings us to uncountable infinite sets.

Resolution of paradoxes

The creation of set theory gave rise to what is considered the third crisis of mathematics, which has not yet been resolved satisfactorily for everyone.
Historically, the first approach was set-theoretic. It was based on the use of actual infinity, when it was considered that any infinite sequence is completed in infinity. The idea was that in set theory one often had to operate on sets that could be parts of other, larger sets. Successful actions in this case were possible only in one case: the given sets (finite and infinite) are completed. A certain success was evident: Zermelo-Fraenkel's axiomatic set theory, a whole school of mathematics by Nicolas Bourbaki, which has existed for more than half a century and still causes a lot of criticism.

Logicism was an attempt to reduce all known mathematics to the terms of arithmetic, and then to reduce the terms of arithmetic to concepts mathematical logic. Frege took up this closely, but after finishing work on the work, he was forced to point out his inconsistency, after Russell pointed out the contradictions in the theory. The same Russell, as mentioned earlier, tried to eliminate the use of impredicative definitions with the help of "type theory". However, his concepts of set and infinity, as well as the axiom of reducibility, turned out to be illogical. The main problem was that the qualitative differences between formal and mathematical logic were not taken into account, as well as the presence of superfluous concepts, including those of an intuitive nature.
As a result, the theory of logicism could not eliminate the dialectical contradictions of the paradoxes associated with infinity. There were only principles and methods that made it possible to get rid of at least non-predicative definitions. In his own reasoning, Russell was Cantor's heir.

At the end of XIX - beginning of XX century. the spread of the formalist point of view on mathematics was associated with the development of the axiomatic method and the program of substantiation of mathematics, which was put forward by D. Hilbert. The importance of this fact is indicated by the fact that the first of the twenty-three problems he presented to the mathematical community was the problem of infinity. Formalization was necessary to prove the consistency of classical mathematics, "while excluding all metaphysics from it." Given the means and methods used by Hilbert, his goal turned out to be fundamentally impossible, but his program had a huge impact on the entire subsequent development of the foundations of mathematics. Hilbert worked on this problem for a long time, having first constructed the axiomatics of geometry. Since the solution of the problem turned out to be quite successful, he decided to apply the axiomatic method to the theory of natural numbers. Here is what he wrote in connection with this: “I pursue important goal: it is I who would like to deal with the questions of the foundation of mathematics as such, turning every mathematical statement into a strictly derivable formula. At the same time, it was planned to get rid of infinity by reducing it to a certain finite number of operations. To do this, he turned to physics with its atomism, in order to show the whole inconsistency of infinite quantities. In fact, Hilbert raised the question of the relationship between theory and objective reality.

More or less full view Finite methods are given by Hilbert's student J. Herbran. By finite reasoning, he understands such reasoning that satisfies the following conditions: logical paradoxes"- only a finite and definite number of objects and functions is always considered;

Functions have precise definition, and this definition allows us to calculate their value;

It never asserts "This object exists" unless a way to construct it is known;

The set of all objects X of any infinite collection is never considered;

If it is known that some reasoning or theorem is true for all these X, then this means that this general reasoning can be repeated for each specific X, and this general reasoning itself should be considered only as a model for such specific reasoning.

However, at the time of the last publication in this area, Gödel had already received his results, in essence he again discovered and approved the presence of dialectics in the process of cognition. In essence, the further development of mathematics demonstrated the failure of Hilbert's program.

What exactly did Gödel prove? There are three main results:

1. Gödel showed the impossibility of a mathematical proof of the consistency of any system large enough to include all arithmetic, a proof that would not use any other rules of inference than those found in the system itself. Such a proof, which uses a more powerful inference rule, may be useful. But if these rules of inference are stronger than the logical means of arithmetic calculus, then there will be no confidence in the consistency of the assumptions used in the proof. In any case, if the methods used are not finitist, then Hilbert's program will turn out to be impracticable. Gödel just shows the inconsistency of calculations for finding a finitist proof of the consistency of arithmetic.
2. Godel pointed out the fundamental limitations of the possibilities of the axiomatic method: the Principia Mathematica system, like any other system with which arithmetic is built, is essentially incomplete, i.e. for any consistent system of arithmetic axioms there are true arithmetic sentences that are not derived from the axioms this system.
3. Gödel's theorem shows that no extension of an arithmetic system can make it complete, and even if we fill it with an infinite set of axioms, then in the new system there will always be true, but not deducible by means of this system, positions. The axiomatic approach to the arithmetic of natural numbers is not able to cover the whole area of ​​true arithmetic propositions, and what we mean by the process of mathematical proof is not limited to the use of the axiomatic method. After Godel's theorem, it became meaningless to expect that the concept of a convincing mathematical proof could be given once and for all delineated forms.

The latest in this series of attempts to explain set theory was intuitionism.

He went through a number of stages in his evolution - semi-intuitionism, intuitionism proper, ultra-intuitionism. At different stages, mathematicians were worried about different problems, but one of the main problems of mathematics is the problem of infinity. The mathematical concepts of infinity and continuity have been the subject of philosophical analysis since their inception (ideas of atomists, aporias of Zeno of Elea, infinitesimal methods in antiquity, infinitesimal calculus in modern times, etc.). The greatest controversy was caused by the use of various types of infinity (potential, actual) as mathematical objects and their interpretation. All these problems, in our opinion, were generated by a deeper problem - the role of the subject in scientific knowledge. The fact is that the state of crisis in mathematics is generated by the epistemological uncertainty of the comparison of the world of the object (infinity) and the world of the subject. The mathematician as a subject has the possibility of choosing the means of cognition - either potential or actual infinity. The use of potential infinity as a becoming, gives him the opportunity to carry out, to construct an infinite set of constructions that can be built on top of finite ones, without having a finite step, without completing the construction, it is only possible. The use of actual infinity gives him the opportunity to work with infinity as already realizable, completed in its construction, as actually given at the same time.

At the stage of semi-intuitionism, the problem of infinity was not yet independent, but was woven into the problem of constructing mathematical objects and ways to justify it. The semi-intuitionism of A. Poincaré and the representatives of the Parisian school of the theory of functions Baire, Lebesgue and Borel was directed against the acceptance of the axiom of free choice, with the help of which Zermelo's theorem is proved, stating that any set can be made completely ordered, but without indicating a theoretical way to determine the elements of any subset of the desired sets. There is no way to construct a mathematical object, and there is no mathematical object itself. Mathematicians believed that the presence or absence of a theoretical method for constructing a sequence of objects of study can serve as the basis for substantiating or refuting this axiom. In the Russian version, the semi-intuitionistic concept in the philosophical foundations of mathematics was developed in such a direction as the effectivism developed by N.N. Luzin. Effectiveism is an opposition to the main abstractions of Cantor's doctrine of the infinite - actuality, choice, transfinite induction, etc.

For effectivism, the abstraction of potential feasibility is epistemologically more valuable than the abstraction of actual infinity. Thanks to this, it becomes possible to introduce the concept of transfinite ordinals (infinite ordinal numbers) on the basis of the effective concept of the growth of functions. The epistemological setting of effectiveism for displaying the continuous (continuum) was based on discrete means (arithmetic) and the descriptive theory of sets (functions) created by N.N. Luzin. The intuitionism of the Dutchman L. E. Ya. Brouwer, G. Weyl, A. Heyting sees freely emerging sequences of various types as a traditional object of study. At this stage, solving mathematical problems proper, including the restructuring of all mathematics on a new basis, intuitionists raised the philosophical question of the role of a mathematician as a cognizing subject. What is his position, where he is more free and active in choosing the means of cognition? Intuitionists were the first (and at the stage of semi-intuitionism) to criticize the concept of actual infinity, Cantor's theory of sets, seeing in it the infringement of the subject's ability to influence the process of scientific search for a solution to a constructive problem. In the case of using potential infinity, the subject does not deceive himself, since for him the idea of ​​potential infinity is intuitively much clearer than the idea of ​​actual infinity. For an intuitionist, an object is considered to exist if it is given directly to a mathematician or if the method of constructing it is known. In any case, the subject can begin the process of completing the construction of a number of elements of his set. The unconstructed object does not exist for intuitionists. At the same time, the subject working with actual infinity will be deprived of this opportunity and will feel the double vulnerability of the adopted position:

1) it is never possible to carry out this infinite construction;
2) he decides to operate with actual infinity as with a finite object, and in this case loses his specificity of the concept of infinity. Intuitionism consciously limits the possibilities of a mathematician by the fact that he can construct mathematical objects exclusively by means that, although obtained with the help of abstract concepts, are effective, convincing, provable, functionally constructive precisely practically and are themselves intuitively clear as constructions, constructions, the reliability of which in practice, there is no doubt. Intuitionism, relying on the concept of potential infinity and constructive research methods, deals with the mathematics of becoming, set theory refers to the mathematics of being.

For the intuitionist Brouwer, as a representative of mathematical empiricism, logic is secondary; he criticizes it and the law of the excluded middle.

In his partly mystical works, he does not deny the existence of infinity, but does not allow its actualization, only potentialization. The main thing for him is the interpretation and justification of practically used logical means and mathematical reasoning. The restriction adopted by intuitionists overcomes the uncertainty of the use of the concept of infinity in mathematics and expresses the desire to overcome the crisis in the foundation of mathematics.

Ultra-intuitionism (A.N. Kolmogorov, A.A. Markov and others) is the last stage in the development of intuitionism, at which its main ideas are modernized, significantly supplemented and transformed, without changing its essence, but overcoming shortcomings and strengthening positive aspects, guided by the criteria mathematical rigor. The weakness of the intuitionist approach was a narrow understanding of the role of intuition as the only source of justification for the correctness and effectiveness of mathematical methods. Taking “intuitive clarity” as a criterion of truth in mathematics, intuitionists methodologically impoverished the possibilities of a mathematician as a subject of knowledge, reduced his activity only to mental operations based on intuition and did not include practice in the process of mathematical knowledge. The ultra-intuitionistic program of substantiating mathematics is a Russian priority. Therefore, domestic mathematicians, overcoming the limitations of intuitionism, adopted the effective methodology of materialistic dialectics, recognizing human practice as a source of formation of both mathematical concepts and mathematical methods (inferences, constructions). The ultraintuitionists solved the problem of the existence of mathematical objects, relying not on the undefined subjective concept of intuition, but on mathematical practice and a specific mechanism for constructing a mathematical object - an algorithm expressed by a computable, recursive function.

Ultra-intuitionism enhances the advantages of intuitionism, which consist in the possibility of ordering and generalizing the methods for solving constructive problems used by mathematicians of any direction. Therefore, intuitionism of the last stage (ultraintuitionism) is close to constructivism in mathematics. In the epistemological aspect, the main ideas and principles of ultraintuitionism are as follows: criticism of the classical axiomatics of logic; the use and significant strengthening (on the explicit instructions of A.A. Markov) of the role of abstraction of identification (mental abstraction from the dissimilar properties of objects and the simultaneous isolation common properties objects) as a way of constructing and constructive understanding of abstract concepts, mathematical judgments; proof of the consistency of consistent theories. AT formal aspect the use of abstraction of identification is justified by its three properties (axioms) of equality - reflexivity, transitivity and symmetry.

To solve the main contradiction in mathematics on the problem of infinity, which gave rise to a crisis of its foundations, at the stage of ultra-intuitionism in the works of A.N. Kolmogorov suggested ways out of the crisis by solving the problem of relations between classical and intuitionistic logic, classical and intuitionistic mathematics. Brouwer's intuitionism as a whole denied logic, but since any mathematician cannot do without logic, the practice of logical reasoning was still preserved in intuitionism, some principles of classical logic were allowed, which had axiomatics as its basis. S.K. Kleene, R. Wesley even note that intuitionistic mathematics can be described as a kind of calculus, and calculus is a way of organizing mathematical knowledge on the basis of logic, formalization and its form - algorithmization. A new version of the relationship between logic and mathematics within the framework of intuitionistic requirements for intuitive clarity of judgments, especially those that included negation, A.N. Kolmogorov proposed as follows: he presented intuitionistic logic, closely related to intuitionistic mathematics, in the form of an axiomatic implicative minimal calculus of propositions and predicates. Thus, the scientist presented a new model of mathematical knowledge, overcoming the limitations of intuitionism in recognizing only intuition as a means of cognition and the limitations of logicism, which absolutizes the possibilities of logic in mathematics. This position made it possible to demonstrate in mathematical form the synthesis of the intuitive and the logical as the basis of flexible rationality and its constructive effectiveness.

Findings. Thus, the epistemological aspect of mathematical knowledge allows us to evaluate revolutionary changes at the stage of the crisis of the foundations of mathematics on turn of XIX-XX centuries from new positions in understanding the process of cognition, the nature and role of the subject in it. Gnoseological subject traditional theory knowledge, corresponding to the period of domination of the set-theoretic approach in mathematics, is an abstract, incomplete, “partial” subject, represented in subject-object relations, torn off by abstractions, logic, formalism from reality, rationally, theoretically knowing its object and understood as a mirror, accurately reflecting and replicating reality. In fact, the subject was excluded from cognition as a real process and result of interaction with the object. The entry of intuitionism into the arena of the struggle of philosophical trends in mathematics led to a new understanding of the mathematician as a subject of knowledge - a person who knows, whose philosophical abstraction must be built, as it were, anew. The mathematician appeared as an empirical subject, already understood as an integral real person, including all those properties that were abstracted from in the epistemological subject - empirical concreteness, variability, historicity; it is an acting and cognizing in real cognition, a creative, intuitive, inventive subject. The philosophy of intuitionistic mathematics has become the basis, the foundation of the modern epistemological paradigm, built on the concept of flexible rationality, in which a person is an integral (holistic) subject of cognition, possessing new cognitive qualities, methods, procedures; he synthesizes his abstract-epistemological and logical-methodological nature and form, and at the same time receives an existential-anthropological and "historical-metaphysical" comprehension.

An important point is also intuition in cognition and, in particular, in the formation of mathematical concepts. Again, there is a struggle with philosophy, attempts to exclude the law of the excluded middle, as having no meaning in mathematics and coming into it from philosophy. However, the presence of an excessive emphasis on intuition and the lack of clear mathematical justifications did not allow transferring mathematics to a solid foundation.

However, after the appearance in the 1930s strict concept algorithm, the baton from intuitionism was taken over by mathematical constructivism, whose representatives made a significant contribution to the modern theory of computability. In addition, in the 1970s and 1980s, significant connections were discovered between some of the ideas of the intuitionists (even those that previously seemed absurd) and the mathematical theory of topos. The mathematics found in some topoi is very similar to that which the intuitionists were trying to create.

As a result, one can make a statement: most of the above paradoxes simply do not exist in the theory of sets with self-ownership. Is such an approach final - controversial issue, further work in this area will show.

Conclusion

Dialectical-materialistic analysis shows that paradoxes are a consequence of the dichotomy of language and thinking, an expression of deep dialectical (Gödel's theorem made it possible to manifest dialectics in the process of cognition) and epistemological difficulties associated with the concepts of an object and subject area in formal logic, a set (class) in logic and set theory, with the use of the abstraction principle, which allows introducing new (abstract) objects (infinity), with methods for defining abstract objects in science, etc. Therefore, a universal way to eliminate all paradoxes cannot be given.

Whether the third crisis of mathematics is over (because it was in a causal relationship with paradoxes; now paradoxes are an integral part) - opinions differ here, although formally known paradoxes were eliminated by 1907. However, now in mathematics there are other circumstances that can be considered either crisis or foreshadowing a crisis (for example, the absence of a rigorous justification for the path integral).

As for paradoxes, the well-known liar paradox played a very important role in mathematics, as well as a whole series of paradoxes in the so-called naive (preceding axiomatic) set theory that caused a crisis of foundations (one of these paradoxes played a fatal role in the life of G. Frege) . But, perhaps, one of the most underestimated phenomena in modern mathematics, which can be called both paradoxical and crisis, is Paul Cohen's solution in 1963 of Hilbert's first problem. More precisely, not the very fact of the decision, but the nature of this decision.

Literature

1. Georg Cantor. Beiträge zur begründung der transfiniten mengenlehre. Mathematische Annalen, 46:481-512, 1895.
2. I.N. Burova. Paradoxes of set theory and dialectics. Science, 1976.
3. M.D. Potter. Set theory and its philosophy: a critical introduction. Oxford University Press, Incorporated, 2004.
4. Zhukov N.I. Philosophical foundations of mathematics. Minsk: Universitetskoe, 1990.
5. Feynman R.F., S. Ilyin. Of course, you are joking, Mr. Feynman!: the adventures of an amazing man, told by him to R. Layton. Hummingbird, 2008.
6. O. M. Mizhevich. Two Ways to Overcome Paradoxes in G. Kantor's Set Theory. Logical and Philosophical Studies, (3):279--299, 2005.
7. S. I. Masalova. PHILOSOPHY OF INTUITIONIST MATHEMATICS. Bulletin of DSTU, (4), 2006.
8. Chechulin V.L. Theory of sets with self-ownership (foundations and some applications). Perm. state un-t. – Perm, 2012.
9. S. N. Tronin. A brief abstract of lectures on the discipline ""Philosophy of Mathematics"". Kazan, 2012.
10. Grishin V.N., Bochvar D.A. Studies in set theory and non-classical logics. Science, 1976.
11. Hofstadter D. Gödel, Escher, Bach: this endless garland. Bahrakh-M, 2001.
12. Kabakov F.A., Mendelson E. Introduction to mathematical logic. Publishing house "Nauka", 1976.
13. YES. Bochvar. On the question of the paradoxes of mathematical logic and set theory. Mathematical Collection, 57(3):369--384, 1944.

Instead of an annotation:

"... Cantor's diagonal proof is an activity for idiots that has nothing to do with what is usually called deduction in classical logic."

L. Wittgenstein

“... Cantor's theory presents a pathological incident in history mathematics, from which the coming generations will simply be horrified"

K. Bauer, founder of topology

1. The crisis of modern mathematical knowledge.

Mathematics plays a leading role in the process of transforming ancient and medieval nuka into modern European one, since theoretical natural science is impossible without mathematics. In modern European natural science, it is no accident that mathematics is called the "queen of sciences." If in the era of antiquity it was separated from the sciences of nature and its subject was the sphere of ideal mathematical entities, then in modern times the situation changes dramatically. Mathematics is approaching the sciences of nature and begins to dictate their own rules of coexistence to them. In this regard, modern conceptual natural science receives the definition of mathematical. Modern natural sciences owe much of their success to modern European mathematics. However, the last, third crisis, which has been going on for more than a hundred years, indicates the existence of serious problems in its foundations.

There is a traditional point of view that at the turn of the XIX-XX centuries. there was a 3rd crisis in the foundations of mathematics, the causes of which are associated with the convergence of mathematics with logic, as well as with the need to clarify such mathematical concepts as number, set, limit, functions, etc.

The origins of this crisis go back to the 17th-18th centuries, when mathematics developed methods for solving problems in natural science. The mathematicians of that time did not particularly care about the rationale for their own methods [L.S. Freynman. Creators of higher mathematics. M., 1968. S. 83-84]

In the 19th century there is a revision of fundamental concepts and the formation of theoretical mathematics. This leads to the formation of set theory and the arithmetization of mathematics.

The greatest mathematicians of the nineteenth century sought to reduce all the facts of mathematics to number and intensively develop, starting with Gauss' Arithmetical Investigations (1801), the theory of number [F.A. Medvedev. The development of set theory in the 19th century. M., 1965. S. 35-36.]. First of all, it applied to mathematical analysis. The most problematic were its logical foundations. In this regard, in the XIX century. the development of the foundations of mathematics and more rigorous methods for its definitions and proofs begins.

In the process of restructuring mathematical analysis, there is a conviction that the theorems of algebra and mathematical analysis can be formulated as a theorem on natural numbers [Dedekind R. What are numbers and what they serve. Kazan: Ed. Imperial University, 1905. S. 5].

The result of this process was the realization of number as a fundamental concept of all mathematics and the construction of the theory of real numbers by such mathematicians as Bolzano, Weierstrass, Dedekind and Kantor.

In the second half of the 19th century, the problem of substantiating mathematics already arises. An outstanding role in its solution was played by the construction of the theory of sets by G. Kantor. As a result, the concepts of analysis and function theory are formulated in terms of set theory. The fundamental concept for the latter was the concept of an actually infinite set.

The development of set theory by incorporating the concept of actual infinity meant, in fact, a revolution in the history of mathematics, comparable to the revolution of Copernicus, the theory of relativity and quantum mechanics. Set theory gave a universal method, which became the basis further development mathematics.

The next stage in the development of mathematics was associated with the convergence of algebra, logic and set theory. Mathematics takes on an unprecedented abstract form. This meant a transition to the logical basis of mathematics. An outstanding contribution to the foundations of mathematics was made by G. Frege (“Fundamentals of Arithmetic” and “Basic Laws of Arithmetic Obtained by Calculus of Concepts”). It carries out the axiomatic deductive construction of mathematical logic (propositional calculus, predicate calculus). The problem of logical substantiation of the number, independence, consistency and completeness of systems of axioms is being solved. “Logistics” arises as a presentation of mathematics in the language of logic. There is a process of development of a powerful logical analysis and formalization of logic.

The idea of ​​the deducibility of mathematics from logic is gaining ground. Frege, having defined the concepts of "number" and "quantity" in the logical terms of "class" and "relationship", manages to formalize the theory of sets, and present mathematics as an extension of logic.

This process ends with the creation of the fundamental three-volume work Principia Mathematica (1910-1913) by Russell and Whitehead.

At the end of the 19th century, a situation in mathematics was very similar to that in physics by the beginning of the 1990s, when the idea of ​​the completeness of classical physics was established. And then followed the dramatic events, on which we dwelled earlier.

At the turn of the XIX-XX centuries. mathematics enters a period of acute crisis caused by the emergence of a series of unsolvable mathematical, logical and semantic paradoxes that cast doubt on Cantor's theory of sets and the foundations of classical mathematics. This plunged even such prominent mathematicians as Cantor, Frege, and others into despair. G. Weil, even after many years, wrote the following lines about this period in the history of mathematical knowledge: “ We are now less than ever sure of the primary foundations of mathematics and logic. We are experiencing our "crisis" in the same way as everyone and everything in the modern world is experiencing it. This crisis has been going on for fifty years already (these lines were written in 1946). At first glance, it seems that it does not particularly interfere with our daily work. However, I must immediately confess that my mathematical work this crisis had a notable practical impact: it directed my interests into areas that I considered relatively "safe", and constantly undermined the enthusiasm and determination with which I pursued my research. My experience was probably shared by other mathematicians who are not indifferent to what place their own scientific activity occupies in this world in the general context of being a person who is interested, suffers and creates.» [M. Kline. Mathematics. Loss of certainty. M.: Mir, 1984. S. 387]. “... The state in which we are now with regard to paradoxes,” writes D. Gilbert, “on long time unbearable. Think: in mathematics - that model of certainty and truth - the formation of concepts and the course of inferences, as anyone studies, teaches and applies them, leads to absurdity. Where to look for reliability and truth, if even mathematical thinking itself misfires? [D. Gilbert. Foundations of geometry. M.-L., 1948. S.349].

Unsuccessful attempts to resolve paradoxes led mathematicians to believe that the causes of the crisis lie in the field of fundamental concepts and methods of reasoning. There is a need to rethink the principles of mathematics and abandon some of the old concepts. And this, first of all, concerned the restructuring of the theory of sets and the refinement of the very concept of a set on a completely new basis [S. Kleene. Introduction to metamathematics. M., 1957. S. 42.]. The very ideal of logic as a criterion for the rigor of mathematical proof was destroyed. Therefore, mathematics faced the task of restoring the former reliability and reliability of mathematical knowledge. The intuitive nature of logical reasoning and the corresponding language no longer suited scientists [Kh. Curry. Foundations of mathematical logic. M., 1969. S. 26.]. Three research programs emerge: logicism, formalism and intuitionism.

A brief digression into the history of modern mathematics shows that at its foundation, and, consequently, the entire mathematical natural sciences lies fundamental theory Cantor sets with the basic scientific concept of actual infinity. And mathematics itself is so closely connected with the concept of infinity that it is often defined as the science of the infinite.

Mathematics, like other sciences (and philosophy), is quite deeply determined by fundamental spiritual and historical paradigms. This belief is confirmed by the works of P.P. Gaidenko, devoted to the evolution of the concept of science in the context of the history of philosophy [P.P. Gaidenko. The evolution of the concept of science (formation and development of the first scientific scientific programs). M. "Science", 1980. – (without footnotes) – [ Electronic resource]. URL: http://www.philosophy.ru/library/gaid/pgaid_physics.html]. And although in his research the author focuses on the interaction of scientific and philosophical knowledge, nevertheless, the impact of the religious context on scientific programs can be traced in them no less clearly. The influence of religious, theological premises on the content of modern mathematics is also convincingly presented in the works of V.N. Katasonova [V.N. Katasonov. Scientific and philosophical concepts of infinity and Christianity. - [Electronic resource]. URL: http://www.bestreferat.ru/referat-73817.html] and A.A. Zenkin [A.A. Zenkin. Transfinite Paradise by Georg Cantor: Bible stories on the eve of the Apocalypse. - [Electronic resource]. URL: http://www.com2com.ru/alexzen/] etc.

Thus, the idea that mathematics is a free (independent) and universal science that develops according to its own laws is greatly exaggerated.

2. Summary of G. Kantor's theory of sets.

G. Kantor considers the Pythagorean-Platonic scientific program as the basis of the theory of sets, the criticism of which was given by Aristotle, but which is again revived in the philosophy of the Renaissance. To substantiate it, theological arguments of the Catholic teaching are used. Philosophical and mathematical thought, starting from the 15th century, gradually prepared the creation of this theory.

Georg Cantor is the creator of set theory and the theory of transfinite numbers. The main idea of ​​his theory of infinite sets consisted in a decisive rejection of Aristotle's thesis about actually infinite sets. Kantor based his study of infinite sets on the idea of ​​a one-to-one correspondence between the elements of compared sets. If such a correspondence can be established between the elements of two sets, then the sets are said to have the same cardinality, that is, they are equivalent or equivalent. "In the case of finite sets," Kantor wrote, "the cardinality is the same as the number of elements." That is why the power is also called the cardinal (quantitative) number of a given set [P. Stakhov. Under the sign of the "Golden Section": Confession of the son of a student student. Chapter 5. Algorithmic theory of measurement. 5.5. The problem of infinity in mathematics. - [Electronic resource]. URL: http://www.trinitas.ru/rus/doc/0232/100a/02320046.htm ].

In 1874, he established the existence of non-equivalent, that is, infinite sets with different cardinalities; in 1878, he introduced general concept cardinalities of sets (in the designation of cardinalities of sets proposed by him and accepted in mathematics by the letters of the Hebrew alphabet, his Jewish origin, according to his father, probably affected). In the main work “On infinite linear point formations” (1879–84), Kantor systematically expounded the doctrine of sets and completed it by constructing an example of a perfect set (the so-called Cantor set) [Kantor G. On infinite linear point formations. // New ideas in mathematics, 1994, No. 6, St. Petersburg].

Kantor gave mathematical content to the idea of ​​actual infinity. Kantor thought of his theory as a completely new calculus of the infinite, "transfinite" (that is, "superfinite") mathematics. Actual infinity is, as it were, a “receptacle” in which a series of potential infinity unfolds, and this receptacle must already be actual data.

According to his idea, the creation of such a calculus was supposed to revolutionize not only mathematics, but also metaphysics and theology, which interested Cantor almost more than scientific research itself. He was the only mathematician and philosopher who believed that the actual infinity not only exists, but is also comprehensible by man in the full sense, and this comprehension will raise mathematicians, and after them theologians, higher and closer to God.. He devoted his life to this task. The scientist firmly believed that he was chosen by God to make a great revolution in science, and this belief was supported by mystical visions.

This approach led Cantor to many paradoxical discoveries that sharply contradict our intuition. So, unlike finite sets, which are subject to the Euclidean axiom "The whole is greater than the part", infinite sets do not obey this axiom. It is easy, for example, to establish the equivalence of the set of natural numbers and its part - the set of even numbers by establishing the following one-to-one correspondence: [P. Stakhov. Under the sign of the "Golden Section": Confession of the son of a student student. Chapter 5. Algorithmic theory of measurement. 5.5. The problem of infinity in mathematics. - [Electronic resource]. URL: http://www.trinitas.ru/rus/doc/0232/100a/02320046.htm].

A set, according to Cantor, is called infinite if it is equivalent to one of its subsets. A set is called finite if it is not equivalent to any of its subsets. A countable set is a set equivalent to the set of natural numbers, since its elements can be enumerated [Ibid.].

Kantor believed that the sets of natural, rational and algebraic numbers have the same cardinality, i.e. are countable [Ibid.].

Cantor also tried to prove that the set N of natural numbers can be mapped onto a part of the set R of real numbers, while the cardinality of the real numbers is greater than the cardinality of the set of natural numbers [Ibid.].

In 1886 Kantor sought to prove that there are no more points in a unit square than in a unit segment. Therefore, the power of the two-dimensional continuum is equal to the power of the continuum of one dimension [Ibid.].

Cantor's ideas turned out to be so unexpected and counterintuitive that the famous French mathematician Henri Poincaré called the theory of transfinite numbers a "disease" from which mathematics must someday be cured. Leopold Kronecker - Kantor's teacher and one of the most respected mathematicians in Germany - even attacked Kantor personally, calling him a "charlatan", "renegade" and "molester of youth" [In the world of science. Scientific American · Russian Edition No. 8 · August 1983 · P. 76–86 / Georg Cantor and the Birth of Transfinite Set Theory].

Set theory also opened a new page in the study of the foundations of mathematics - Kantor's work made it possible for the first time to clearly formulate modern general ideas about the subject of mathematics, the structure of mathematical theories, the role of axiomatics and the concept of isomorphism of systems of objects, given together with the relations connecting them. His set theory is one of the cornerstones of mathematics.

In the philosophy of mathematics, Kantor analyzed the problem of infinity. Distinguishing two types of mathematical infinite - improper (potential) and proper (actual, understood as a complete whole), - Kantor, unlike his predecessors, insisted on the legality of operating in mathematics with the concept of actually infinite. A supporter of Platonism, Kantor saw in the mathematical actual-infinite one of the forms of the actual infinite in general, acquiring the highest completeness in the absolute Divine being.

3. The great confrontation between Cantorians and anti-Cantorians.

Criticism by A.A. Zenkin of abstract set theory

G. Kantor and "Teachings about the Transfinite".

Among the numerous critical literature devoted to G. Kantor's set theory, the studies of the Russian mathematician A. A. Zenkin deserve special attention. According to the famous mathematician A.P. Stakhov, perhaps it is he (Zenkin) who will put the last point in the dispute with Kantor and in resolving the mathematical crisis in modern mathematics[ http://www.trinitas.ru/rus/doc/0232/100a/02320046.htm].

In the original article “George Kantor's Transfinite Paradise. Biblical stories on the threshold of the Apocalypse "Russian scientist A.A. Zenkin analyzes the epistemological flaws in the logic of Cantor's proof of the uncountability of the continuum, based on the concept of actual infinity[A.A. Zenkin. Transfinite paradise of Georg Kantor: Biblical stories on the threshold of the Apocalypse. - [Electronic resource]. URL: http://www.com2com.ru/alexzen/].

For millennia, - notes A.A. Zenkin, - such outstanding scientists and philosophers as Aristotle, Euclid, Leibniz, Berkeley, Locke, Descartes, Kant, Spinoza, Lagrange, Gauss, Kronecker, Lobachevsky supported and shared a negative attitude towards the concept of AB , Cauchy, F. Klein, Hermite, Poincare, Baer, ​​Borel, Brouwer, Quine, Wittgenstein, Weil, Luzin, and already today - Erret Bishop, Solomon Feferman, Yaroslav Peregrin, Vladimir Turchin, Pyotr Vopenka and many others.

Since the 70s of the 19th century, there has been a sharply negative attitude towards the theory of sets by Georg Cantor, based on the concept of AB. A.A. Zenkin gives examples of the most categorical statements addressed to her. So, Henri Poincaré came to the conclusion that “there is no actual infinity; the Cantorians forgot about it and fell into controversy. Future generations will view Cantor's set theory as a disease that has finally been cured."[A.Poincaré, On Science. – M.: Nauka, 1983]. The founder of modern topology, L. Brouwer, is no less radical in his statements: “ Cantor's theory as a whole is a pathological incident in the history of mathematics, from which future generations will simply be horrified.[A.A. Frenkel, I. Bar-Hillel. Foundations of set theory. - M.: "Mir"].

“Nevertheless, even today,” the Russian mathematician writes, “just like at the beginning of the 20th century, there is a “great confrontation” between the meta-mathematical logic of the Cantorians, who recognize the legitimacy of Cantor’s “Doctrine of the Transfinite” in the form of “non-naive” (see. below) versions of this "Teaching", i.e. in the form of modern axiomatic set theory (hereinafter - ATM), based on the (tacit - see below) use of the concept of AB, and the mathematical intuition of anti-Cantorians who reject the concept of AB and G. Cantor's "Doctrine of the Transfinite" based on this concept" [ A.A. Zenkin. Transfinite paradise of Georg Kantor: Biblical stories on the threshold of the Apocalypse. - [Electronic resource]. URL: http://www.com2com.ru/alexzen/].

The use of the AB concept leads to paradoxes of logic and mathematics, the generation mechanisms of which remain undiscovered until now. In this regard, the disclosure of the logical nature of paradoxes and the legitimacy of using the AB concept in mathematics are relevant today. Frenkel and Bahr-Hillel point out that there is absolutely nothing in the traditional interpretation of logic and mathematics that could serve as a basis for eliminating Russell's antinomy.<АЗ: а также парадокса «Лжец»>. We believe that any attempts to get out of the situation with the help of traditional ... ways of thinking, so far invariably failed, are obviously insufficient for this purpose. Some departure from the usual ways of thinking is clearly necessary, although the place of this departure is not clear in advance” [A.A. Frenkel, I. Bar-Hillel. Foundations of set theory. - M.: "Mir"].

Abstract set theory and its assertion in modern science, according to A.A. Zenkin, is a vivid example of pseudo-science, an unprecedented case of creating a false myth in science through the use of PR technologies.

Moreover, A.A. Zenkin involuntarily reveals the true impartial essence of modern science as a social institution: “ATM - the initiative gave rise to such a large-scale negative phenomenon as bourbakism, i.e. excessive, unnecessary, meaningless, stupefying, stupefying and zombifying formalization of mathematics and mathematical education. Describing the negative consequences of such burbakization, an outstanding Russian mathematician and teacher, academician V.I. Arnold writes: “In the middle of the 20th century, the mafia of “left-hemispheric mathematicians”, which had great influence, managed to exclude geometry from mathematical education ... replacing the entire content side of this discipline with training in formal manipulation of abstract concepts. Such an abstract description of mathematics is not suitable for teaching or for any practical applications. Modern formalized (burbakized) education in mathematics - complete opposite teaching the ability to think and the basics of science. It is dangerous for all mankind. The future of mathematics infected with this disease looks rather gloomy” [A.A. Zenkin. Transfinite paradise of Georg Kantor: Biblical stories on the threshold of the Apocalypse. - [Electronic resource]. URL: http://www.com2com.ru/alexzen/].

The Russian mathematician formulates four examples of “lie to save H. Kantor’s ATM”:

Lie first. "Mathematics is the queen of all sciences, and ATM is the queen of mathematics"! On this occasion, A.A. Zenkin writes that the modern ATM fools the professional mathematical community and zombifies the young generation of mathematicians. Cantorians argue that if at the beginning of the 20th century many outstanding mathematicians categorically rejected ATM as a pseudo-science, today, “modern mathematicians, finally, enlightened about it that all infinities are relevant changed their minds on the subject that the theory final natural numbers is "derivable" from the theory transfinite numbers, that the concept of an empty set is deduced from the concept of an actually infinite set, that all modern mathematics can be derived from ATM and officially recognized that "Mathematics is the Queen of all sciences, and ATM is the Queen of Mathematics"! All yesterday's opponents of ATM today agree that ATM is outstanding achievement modern mathematics, an achievement that changed the face of all mathematics in the 20th century” [Ibid.].

“This is an empirical fact,” Martin Davis and Reuben Hersh are already zombifying the scientific community today, “ that about 90% of working mathematicians accepted Cantor's set theory, both in theory and in practice, to some extent» [Ibid.].

Then, as in fact, A.A. Zenkin notes, the Cantorians are deliberately cunning and do not make a significant difference between the language of abstract set theory and Cantor's doctrine of transfinite ordinals and cardinals. Indeed, the language of set theory has become a universal mathematical language. While the doctrine of transfinite ordinals and cardinals, due to their absolute uselessness, 90% of really working mathematicians do not apply anywhere. Of the rest, 9% of mathematicians categorically do not accept this doctrine, and only 1% are ATM-adepts or Bourbakists.

Second lie. The basis of modern ATM is a blatant pseudo-scientific, semi-criminal "method of solving" the fundamental scientific question about the logical nature mathematical infinity . Its essence lies in the fact that Cantor's set theory, based on the concept of AB, was declared "naive", and the term AB itself was taken out of the bounds of a respectable meta-mathematical science. It was one of the most effective PR campaigns ever implemented in the history of science.

Nevertheless, the modern theory of ATM borrowed from the "naive" theory the theorem on the uncountability of the continuum, the proof of which is based on the use of the obviously contradictory concept of AB. In this regard, A.A. Zenkin considers Cantor's theory of sets as one of the main sourcesThe Third Great Crisis of the Foundations of Mathematics, which continues to this day.

Third lie. The conditions for proving ATM are not formulated explicitly, but are implied at the level of philosophical provisions. From the point of view of classical logic and mathematics, the "AB assumption" is a necessary condition for the deduction of most ATM theorems.

Fourth lie. Set theory failed, in the end, to eliminate potentiality by scientific methodology, i.e. prove the inconsistency of the PB concept. ATM went the other way. She declared the problem of the legitimacy of the use of AB as a philosophical one. A.A. Zenkin sees this as the instinct of self-preservation of ATM supporters, since an attempt to give a strict definition of the concept of AB will lead to an obvious understanding of its inconsistency. And this will jeopardize the well-funded and habitual the well-being of ATM regulars in Cantor's "transfinite paradise". In such a semi-criminal and pseudo-scientific way, the ATM - "clan", dealt with its opponents.

And finally, the fifth lie. Imposing a “horror story” on the mathematical community that the proof of the Uncountable Continuum Theorem is so difficult that it is available only to selected professionals . Many mathematicians believed in this myth and admitted their incompetence when discussing Cantor's fundamental theorem about the uncountability of the continuum. As proof of the flagrant falsity of this myth, A.A. Zenkin proposes to compare the methodology of proving the Cantor theorem and the well-known Pythagorean theorem.

In the Pythagorean theorem, notes A.A. Zenkin, three (!) elementary concepts mathematics (concept right triangle, the concept of similarity of triangles, the concept of proportion) and three (!) mathematical operations are performed: two multiplications and one addition algebraic expressions. The proof itself (without the picture) takes 5 (five!) lines. Cantor's proof uses three (!) elementary concepts of mathematics (the concept of a natural number, the concept of a real number, and the concept of an infinite sequence of enumerated real numbers) and not a single (!) mathematical operation is performed. The proof itself takes 5 (five!) lines, written in the language of elementary logic of the second half of the 19th century[Ibid].

The correctness of this proof meets with serious objections from eminent mathematicians, logicians and philosophers. " In its paradigm implications for philosophy, logic, mathematics, and the psychology of knowledge, Cantor's theorem is unparalleled. Such a different epistemological “fate” of these theorems, so similar in formal criteria (and in terms of the “screaming” triviality of the proofs), is explained by the fact that the proof of Cantor’s theorem uses the (implicitly) contradictory concept of actual infinity» [Ibid.].

A.A. Zenkin does not stop at this argument and proceeds directly to the analysis of the diagonal method (DM) as a proof of Cantor's theorem on the uncountability of the continuum.

Considering the canonical form of DM, the Russian scientist comes to the conclusion that “his (Cantor’s) diagonal proof of the quantitative incommensurability of two infinite sets X and N is based on the fact that an infinite set X always contains one extra element (Cantor’s new AD-d.h. x*), for the enumeration of which, "as always", there is one element missing from the infinite set N, or, formally, from the fact that the infinite set X has one more element than the infinite set N. I think this is - it is precisely that place in Cantor's proof that has always caused categorical rejection (rejection) on the part of the scientific intuition of outstanding mathematical professionals (see List-1)” [Ibid.]. A.A. Zenkin gives an assessment of such evidence by Wittgenstein: “A person works day by day in the sweat of his brow - he makes a list of all real numbers, and now, when the list is finally finished, a magician appears, takes the diagonal of this list and in front of his eyes of the astonished audience, with the help of a rather "esoteric" algorithm, turns it into ... an anti-diagonal, i.e. to a new AD-real number that is not contained in the original list . Of such kindCantor's diagonal proof is an activity for idiots that has nothing to do with what is called deduction in classical logic..

Moreover, the Russian mathematician discovers for the first time unique fact in Cantor's proof. The key point of Cantor's proof is the explicit use of the counter-example method. And “the counter-example itself is not found in the set of all possible realizations of a given general assertions, but algorithmically deduced from the general assertion that this counterexample is intended to refute (in the form deductive output , here B= "list (1) contains all d.h. from X”)” [A.A. Zenkin. Transfinite paradise of Georg Kantor: Biblical stories on the threshold of the Apocalypse. - [Electronic resource]. URL: http://www.com2com.ru/alexzen/].

As a result of acquaintance of ATM professionals with the discovery of A.A. Zenkin, a sharp controversy arose, in which “all the fake professionalism of a number of recognized ATM authorities was manifested precisely in the field of elementary logic” [Ibid.].

Summarizing the results of the controversy, A.A. Zenkin comes to the following unexpected conclusion: “A scandalous situation arises! – For more than a hundred years, outstanding (and not so) professionals in the field of meta-mathematics, mathematical logic, axiomatic set theory and other Bourbakists have been teaching (more correctly, zombifying) new generations of students every year, “how to correctly prove” the uncountability of the continuum using the famous diagonal method Cantor, absolutely not understanding the logical nature of this method!

Truly, "a pathological incident from which, according to Brouwer, future generations will be horrified"! - Or, rather, they will laugh "from the depths of their souls", but ... "until I completely fall." - Over whom? - I think about those 90% of "working" mathematicians who for a whole century "completely disinterestedly" conceded their "queen of all sciences" for clearly "misuse" by "left-brain patients". For laughing at the sick, even the left-brained ones, is sinful and pointless.[Ibid].

The Russian mathematician completes the critical analysis of the DMC proof with the story of David Hilbert's dramatic paradox proposed about 80 years ago. In the 1920s, D. Hilbert, in order to demonstrate the fundamental differences between finite and infinite sets in Cantor's set theory, proposed a popular paradox under the name "Grand Hotel". The presentation of the paradox itself is rather cumbersome, so let us formulate its essence. The “Grand Hotel” paradox demonstrates a fundamental property of infinite sets: “... if a finite or countably infinite set is added to an infinite set, then the cardinality of the first set will not change” [Ibid.].

Comparing the DMC proof with D. Hilbert's paradox, A. A. Zenkin comes to a remarkable conclusion: the DMC proof of the uncountability of the continuum is a deductive model (in the sense of Tarski) of D. Hilbert's "Grand Hotel" paradox.

In D. Hilbert's paradox, we are dealing with a potentially infinite process, which has the following fundamental property: until this process ends, “There is no (logical and mathematical) reason to assert that the assumption "X is countable" is false. Therefore, in the event that the set Y 1 is countably infinite, the statement of Cantor's Theorem "X is uncountable" is unprovable"[Ibid].

The above arguments, A.A. Zenkin concludes, indicate that “Cantor's theorem on the uncountability of the continuum is unprovable. This means that the distinction between infinities by the number of elements is myth-making. But if the uncountability of the continuum is unprovable, then G. Cantor's theory of transfinite sets is not just "naive", but frank pseudo-science, and therefore G. Cantor's transfinite "paradise" can be closed without any damage to really "working" mathematics"[Ibid].

Concluding the presentation of A.A. Zenkin’s critical studies on the theory of infinite sets by Georg Kantor, I would like to emphasize the importance of his following conclusion. Cantor's theorem is incorrect from the point of view of Aristotle's classical logic.

4. Criticism of the axiomatic approach of A.A. Zenkin

The axiomatic approach proposed by A. Zenkin for the concept of AB and PB is, from our point of view, methodologically incorrect.

Aristotle's axiom and Cantor's axiom are formulated through the concept of infinity, which is not strictly and formally defined. Based on the formulation of the axioms, it follows that PB and AB are types of the infinite as such, i.e. kind.

Second moment. The concept of PB and AB Aristotle considered on the basis of his own doctrine of being and essence based on the laws of classical logic (traditional). While Cantor, in his theory of sets, proceeded from the Pythagorean-Platonic research program. Plato's doctrine of being and essence is alternative to peripatetic philosophy and is consistent with dialectical logic and the principle of the coincidence of opposites.

Aristotle did not consider the concepts of AB and PB as contradictory, primarily because the concept of infinity is very specific and the principles and laws of traditional logic are inapplicable to it. Aristotle called it an illegitimate concept, which is generally not given to either our feeling or thinking. The Infinite exists only in possibility, not in reality. For if it existed in reality, it would be a certain (certain) quantity, or a finite value. Therefore, the infinite exists as a property.

Infinity, according to Aristotle, is where, taking a certain amount, you can always take something after it. And where there is nothing outside, it is the whole. The infinite is that which is absent from something, being outside of it. "The whole and limited (infinite] is not in itself, but in relation to another; and since it is infinite, it does not embrace, but is embraced. Therefore, it is not cognizable as infinite, because matter [as such] has no form. Thus, it is clear that the infinite fits the definition of a part rather than a whole, since matter is a part of the whole, like copper for a copper statue.If it embraces sensible objects, then in the field of the intelligible "big" and "small" must to embrace the intelligible [ideas], but it is absurd and impossible for the unknowable and indefinite to embrace and define" [Aristotle. Collected works in 4 volumes. V.3, Moscow, "Thought", 1981, p.120 ].

Consequently, Aristotle considers the concept of infinity in close connection with the key categories of his philosophy: form - matter, possibility - reality, part - the whole. In this context, the concept of AB is not contradictory to PB, but completely unthinkable from the point of view of Aristotle's logic. Contradictorial PB is rather the concept of the finite, as the relation of the indefinite and the definite. If PB is considered in the context of a part - a whole, then the definition of a part is more suitable for it. Then in relation to it, the actual infinite more corresponds to the concept of the whole. In this case, PB is a concept subordinate to the concept of AB. This is exactly how G. Kantor himself interpreted it.

Thus, for Aristotle, one can only speak of infinity in the sole sense of PB. A concept cannot be related to it, which is not recognized as a concept, i.e. AB. And the very concept of PB is indefinite, unknowable and has no reality.

It is this special status of the concept of infinity, which Aristotle speaks of, that does not allow us to apply the traditional operations of formal logic to it. The concept of PB is not a mathematical object in the strict sense of the word.

That the concept of infinity does not belong, in the strict sense, to mathematics follows from the definitions of number and magnitude. Here is, once again, the definition of Aristotle. “Quantity is that which is divisible into component parts, each of which, whether there are two or more of them, is by nature something one and a definite something. Every quantity is a set if it is countable, and magnitude if it is measurable. A set is that which, in the possibility, is divisible into parts that are not continuous, a quantity - into parts that are continuous ... Of all these quantities limited set is number limited line length, limited width - flat, limited depth is the body” [Aristotle. Op. in 4 volumes. Volume 1. M.: Thought, 1976, p.164]. From the above passage of Aristotle it follows that the main subject of mathematics is the concept of magnitude and number. The number is a limited set, the value is a limited geometric space (line, plane, body). Unlimited set and unlimited space are infinity, as two forms of quantity, having no boundaries, end or limit. Therefore, they are indefinite, and therefore unknowable.

Furthermore, infinity for Aristotle is a property of thinking, first of all, and not the subject of physics or mathematics. « To trust thinking in the question of the infinite is absurd, since excess and deficiency (in this case) are not in the object, but in thinking. After all, each of us can be mentally imagined many times more than he is, increasing it to infinity, however, not because someone is outside the city or has some size because someone thinks this way, but because it is so [on actually]; and the fact [that someone thinks this way] will be [for him] an accidental circumstance” [Ibid.]. If the infinite does not exist in the object, then what do we axiomatize - the activity of thinking? And what does mathematics have to do with it? For its subject matter is pure quantity: number and magnitude?

The concept of actual infinity Cantor constructs, following the tradition of the Pythagoreans, who, as Aristotle testifies, "composed magnitudes from numbers." Kantor believes that a continuous quantity can be measured by a number as a true set of indivisible units. It is clear that such an approach is completely unacceptable for Aristotle. For him, the value is divided only into divisible parts. Therefore, a quantity cannot be composed of indivisibles. Otherwise, Zeno's aporias about the contradiction of movement will not be resolved, and it will also be impossible to explain the possibility of movement, the continuity of time and space.

According to Kantor's axiom, according to Zenkin, it follows that he denies potential infinity. Kantor not only did not deny PB, but did not consider it at all as actually infinite. For him, PB is a variable finite quantity. Moreover, he believed that if you take PB, then all the more you should take AB.

The conclusion is the following. The axioms of Aristotle and Cantor, formulated by Zenkin, do not reflect the actual attitude to the concept of PB and AB of Aristotle and Cantor. In both axioms, in Aristotle's axiom (4th century BC): "All infinite sets are potentially infinite sets", and in more than a hundred years of the de facto existing and contradictory Cantor's axiom (XIX century AD) : "All infinite sets are actual-infinite sets" [see A.A. Zenkin. Transfinite paradise of Georg Kantor: Biblical stories on the threshold of the Apocalypse. - [Electronic resource]. URL: http://www.com2com.ru/alexzen/ ], the generic concept of "infinite set" is defined through its type. In Aristotle's axiom - through potentially infinite sets, in Cantor's axiom - through actually infinite sets. Neither the concept of PB nor AB are mathematical objects in the strict sense of the word, since they exist only in possibility, are unknowable and indeterminate. The concept of AB and PB is neither a number nor a magnitude, but is a property of our abstract rational thinking.

All of the above has nothing to do with that part of Zenkin's work, in which he proves, on the basis of classical logic, that Cantor's theorem on the uncountability of the continuum is unprovable. Zenkin showed that Cantor's Diagonal Method (DMC), underlying the proof of the theorem, is a specific version of a counter-example well known to Pythagoras and Euclid. And the famous paradox "Grand Hotel" by D. Hilbert is a deductive model (in the sense of A. Tarsky) of the DMC-proof of the uncountability of the continuum by G. Kantor. Based on this model, Zenkin concludes that the DMC proof is incorrect from the point of view of classical logic. Therefore, there are no uncountable sets, and all infinite sets have the same cardinality. Thus, the whole grandiose “Teaching about the Transfinite” by G. Kantor is collapsing.

Thus, the main conclusion that suggests itself upon careful study of the theorem on the uncountability of the continuum and the theory of Cantor's transfinite numbers based on it, is that its falsity is quite easily (as A.A. Zenkin showed) refuted on the basis of Aristotle's classical logic.

And no less important, the last conclusion. Cantor's theory is not an accidental phenomenon in European mathematics, but a natural result of the identification of the concepts of number and magnitude, which led to the gradual arithmetization of mathematics, its speculativeness and immoderate abstractness.

5. The mystery of potential infinity

An equally important question, which Zenkin involuntarily raised when proving the inconsistency of Cantor's theorem on the uncountability of the continuum, is directly related to the essence of potential infinity recognized in mathematics.

In the 1920s, David Hilbert proposed a popular paradox called the "Grand Hotel" (hereinafter, for brevity, GO), which illustrates the fundamental difference between finite and infinite sets in Cantor's (as well as in modern axiomatic) set theory. We will not present the paradox itself, since it is rather cumbersome. Its content is that it very clearly demonstrates the main property of infinite sets: if a finite or countably infinite set is added to an infinite set, then the cardinality of the first set will not change.

Zenkin shows that the DMC proof of the uncountability of the continuum is a deductive model (in the sense of Tarski) of D. Hilbert's GO paradox [A.A. Zenkin. Transfinite paradise of Georg Kantor: Biblical stories on the threshold of the Apocalypse. - [Electronic resource]. URL: http://www.com2com.ru/alexzen/].

Having established that in the DNA method Kantor does not use the AB process, but the PB process, Zenkin notes that no one will know the truth of the assertion of this theorem, since the infinite process does not have the last element.

Zenkin showed that Cantor's actual infinity, being necessary condition of the DMC proof of the uncountability of the continuum, in fact, is potentially- endless discussion. "This proves that" topical" and "infinite" in the framework of Cantor's proof. Theorems on the uncountability of the continuum are (logically and algorithmically) contradictory concepts, and, consequently, the concepts of " topical" and " final» are algorithmically identical» [Ibid.]. And if this is a potentially infinite statement, then its truth cannot be established, since an infinite process does not have a last element. This Zenkin's conclusion confirms our assumption that the notion of finite, and not AB, contradicts the notion of PB.

Thus, Zenkin writes, first proven great intuitive providence (and warning!) of Aristotle, Euclid, Leibniz, and many other (see List-1) outstanding logicians, mathematicians and philosophers that “ actual infinity" is an internally contradictory concept (something like " finished(by Cantor) infinity”) and therefore its use in mathematics is unacceptable” [Ibid.].

Unfortunately, to prove the internal inconsistency of the concept of actual (completed, i.e. completed, finite) infinity is, to a certain extent, vain work, due to its obvious direct inconsistency. Within the framework of classical Aristotelian logic, this is simply impossible. In the context of speculative (dialectical) logic, which denies the law of contradiction, this is quite acceptable.

Zenkin also discovers that the canonical form of Cantor's "diagonal" proof of the uncountable continuum theorem is identical to the canonical infinite form (P2) of the "Liar" paradox:

“Someone says “I am a liar”. - Is he a liar? If he is a liar, then he is lying, claiming that he is a liar; therefore he is not a liar. But if he is not a liar, then he is telling the truth, claiming that he is a liar; therefore, he is a knave, or, in short (here A = "I am a knave"): and [ØA ® A] (P1)" [Ibid.]

Zenkin also notes, “What the simulation of the liar paradox is not to an analog computer proves. That this paradox has not a finite form, but the following infinite A ® ØA ® A ® ØA ® A® ØA ® A ® ... (P2) and there are no logical and mathematical reasons, reasons or grounds for completing this potentially-infinite process” [Ibid.].

As a result, the Russian mathematician makes an interesting conclusion. “It should be emphasized that it is the infinite form (P2) that implements the necessary and sufficient conditions (in the strict logical and mathematical sense) of the very phenomenon of paradoxicality. In this case, the true "semantics" of this paradox is not at all that the statement "I am a liar" "can neither be true nor false", but that this statement, on the contrary, is both both true and false“at the same time, in the same place and in the same respect.” In other words, in the “Liar” paradox in the form (P2), truth and falsehood are mixed-mixed, which means that truth and falsehood become indistinguishable” [Ibid.].

It's hard to disagree with this. According to Plato, the infinite is that which has an indefinitely quantitative characteristic and does not allow a strict definition. He calls the infinite "indefinite duality", it always has two meanings and cannot take on one meaning, cannot be determined.“... the infinite can exist as a day exists or as a competition - in the sense that it becomes always different and different» [P.P. Gaidenko. The history of Greek philosophy in its connection with science. - [Electronic resource]. URL: http://www.philosophy.ru/library/gaid/0.html].

The question arises, what is the logical meaning of the Platonic concept of the infinite as a process of "becoming always different and different"? In our opinion, the very concept of potential infinity implicitly contains a principle that denies the law of contradiction. This “other and other,” instead of “something else,” is the uncertainty principle. If the law of contradiction in the interpretation of Aristotle is formulated as follows: “It is impossible for the same thing to be and not to be inherent in the same thing and in the same sense,” then, in our case with the definition of PB, “one and the same same” is identical in meaning to the concept of “other” in Plato. Therefore, in Plato's definition, we are dealing with a statement that denies the law of contradiction. For example, consider a series of natural numbers: 1, 2, 3, 4, 5… as an example of potential infinity. If we take any pair of neighboring numbers, then it is impossible for all three types of their ratios to be true in magnitude: 3 > 4, 4 > 3, or 3 = 4. If we take a finite number 4, then, for example, in relation to its magnitude, it cannot be greater than itself. Whereas, in an infinite number series, the value of a number is always changing, and we cannot apply the law of contradiction as the law of certainty to it. Therefore, potential infinity is equally inherent in all numbers of the natural series: 1, and other (2), and other (3), and other (4). Therefore, the sign of the disjunction must be replaced by the conjunction. And the introduction of the law coincidentia oppositorum instead of the law of contradiction leads to paradoxes. What is a paradox? This is a contradictory statement.

And finally, an example of the Liar paradox. Someone says, "I'm lying." If he is lying, then what he said is a lie, and therefore he does not lie. If he does not lie, what he says is the truth, and therefore he is lying. In any case, it turns out that he is lying and not lying at the same time [Logical Dictionary-Reference. N.I. Kondakov. The science. M., 1976. S.433]. In this paradox, we are dealing with a deliberate violation of the law of contradiction. It is impossible for someone to lie and not lie in the same respect. And this violation is inherent in the structure of the paradox.

Thus, as Zenkin shows, and this follows from the analysis of this paradox based on classical logic, the violation of the law of contradiction is implicitly inherent in the content of the concept of potential infinity, which leads to the phenomenon of paradox. If we are talking about a series of natural numbers, then each of the natural numbers that make up the series is both included and not included in the infinite series of natural numbers. First, a number, for example 5, enters when we reached it during the calculation, and then, the number 6 changes it, and so on. Certainty is constantly changing, and, therefore, perhaps impossible, the appearance of paradoxes.

If in the concept of AB the inconsistency and paradoxical nature of this concept is obvious, then in the concept of PB it is hidden.

Comprehension of the nature of PB, one cannot ignore the concepts of arithmetic and geometric infinity. Let's consider these concepts in more detail.

The sequence of natural numbers 1, 2, 3, ..., (1)

represents the first and most important example of an infinite set. Since the time of Hegel, the arithmetic infinity of the natural series 1 + 1 + 1 + ..., due to its hopelessness, has been called “bad” or “bad” infinity.

Geometric infinity consists in the unlimited division of the segment in half. Pascal wrote the following about geometric infinity: “There is no geometer who would not believe that space is divisible to infinity. He cannot do without it, just as a man cannot be without a soul. And yet there is no man who understands infinite divisibility…” [ A.P. Stakhov Under the sign of the "Golden Section": Confession of the son of a student battalion. Chapter 5. Algorithmic measurement theory. 5.5. The problem of infinity in mathematics. Potential and actual infinity. - [Electronic resource]. URL: http://www.trinitas.ru/rus/doc/0232/100a/02320046.htm].

Indeed, this is an extremely important question that cannot be resolved within the framework of the currently dominant anthropocentric paradigm.

“The first naive impression produced by natural phenomena and matter,” writes D. Gilbert, “is the impression of something continuous, continual. If we have in front of us a piece of metal or a certain volume of liquid, then the idea is imposed on us that they are indefinitely divisible, that an arbitrarily small piece of them again has the same properties. But wherever the methods of investigation in the physics of matter are sufficiently improved, we come across the limits of this divisibility, which lie not in the imperfection of our experience, but in the nature of the thing itself, so that one could directly perceive the trend of modern science as liberation from the infinite small; now it would be possible to counter the old thesis “natura non facit saltus” (nature does not make leaps) with the antithesis: “nature makes leaps” [Gilbert D. On the Infinite. Scan source: Gilbert D. On the Infinite // Him. Foundations of geometry. - M.-L., 1948. 491 p. (abridged article from Mathematischen Annalen, v. 95.) - [Electronic resource]. URL: http://www.fidel-kastro.ru/matematika/gilbert/hilbert2.htm].

“Infinite divisibility exists only in mathematics. In nature, the experiments of physics and chemistry are nowhere to be found - therefore, this is only a mathematical idea - a product of mathematical thinking! Idea infinite universe dominated for a long time before Kant and after. But this idea is the reverse side of the limitations of our experience and the process of cognition” [Ibid.].

The property of geometric infinity as an unlimited divisibility of a segment in half is unsolvable within the framework of geometry, and requires the involvement of philosophy and theology.

Firstly, the process of segment division expresses the fundamental property of rational thinking - the destruction (division) of the object under study. The understanding acts in a dividing way in relation to its objects, thanks to which certainty is achieved.

Secondly. The infinite divisibility of a segment is due to the fact that a geometric segment is a form of continuous quantity. And quantity itself is an abstraction of sensible things, indifferent to quality.

In the objective material world there is no pure quantity, all things have a measure and thanks to it they are identical to themselves and differ from others. Measure is the direct unity of quality and quantity. In a geometric segment, we are dealing with immensity, i.e. measure going beyond the limits of its qualitative certainty. Any objective thing has the limits of its qualitative existence. If they are destroyed, then the thing itself is destroyed. Therefore, a sensible (finite) thing cannot be divided to the state of potential (bad) infinity. The qualitative definiteness of a thing opposes this process of division. For example, a piece of a tree can be divided as long as the pieces from division retain the properties of this tree, i.e. to the molecular boundary of the cellulose molecule. Further division of the cellulose molecule is a process of division of another thing, therefore, the process of division of wood has a lower limit - cellulose molecules. Molecular division will have a lower limit at the atomic level. Division specific atoms elements will lead to division to the level of subatomic parts, etc. Consequently, any division of objective things is finite. If we consider the process of division without taking into account quality and measure, then the process of division really becomes infinite. But what do we measure then? Abstraction of finite things - matter. Matter as an objective sensible thing (in a natural, native, unaltered reality) does not exist, it is the same product of an abstract thought as the geometric segment itself.

Thus, both the geometric segment and matter are divisible up to a bad (potential) infinity. But here we are not dealing with real sensible things, but with pure quantity, which has no measure in itself, and therefore constantly goes beyond its limits. It is no coincidence that Hegel wrote in the Science of Logic that the concept of quantity contains the need to go beyond its boundaries.

Returning to the definition of Aristotle: “Quantity is that which is divisible into its component parts, each of which, whether there are two or more of them, is by nature one thing and a certain something ...” [Aristotle. Op. in 4 volumes. Volume 1. M .: Thought, 1976, p. 164], it is obvious that mathematics deals with pure quantity, i.e. not with the sensible quantity of finite things that physics studies, but with the abstract pure immeasurable quantity - number and magnitude. Therefore, in sensible nature as a subject of physics, there is not only actual or potential infinity. The world is finite in both the extensive and the intensive sense. Not surprisingly, Aristotle noted that the infinite is not given to either feeling or mind, and called it an illegal concept. God arranged everything in the created world according to measure, number and size (Holy Scripture).

In his hierarchy of forms of knowledge, Aristotle, after metaphysics (by which he in the strict sense understood theology as the science of the eternal) as the first philosophy, put physics, and only then mathematics. And this is absolutely true, since the subject of mathematics - pure quantity, has its roots in sensible material nature. Its subject matter is number and magnitude as forms of abstract quantity. Historical development abstract form in mathematics led to the fact that the main subject of its study was the sphere of ideal mathematical objects: number, magnitude, point, line, set, etc., which largely does not coincide with the world of real physical objects. The concept of potential infinity is one of them. Therefore, the conclusion that suggests itself here is, firstly, that it is necessary to clearly recognize the features and boundaries of mathematics and the natural sciences. And, secondly, in the study of nature (physics, biology, etc.) it is necessary to rely on and proceed from the content of the immediate subject, and not from a priori mathematical models. And although the history of mathematics has many examples of reverse interaction, nevertheless, this practice has many fundamental exceptions.

6. Number theory and set theory G. Kantor

"The subject matter of number theory coincides with

subject (study) of all mathematics.

A.M. Vinogradov

Historically, the formation of the concept of number took place on the basis of a formal operation of generalization (expansion) of the volume due to the inclusion of new types of numbers (sets) in its composition.

The first ideas about the number arose from counting people, animals, fruits, various products, etc. The result is natural numbers: 1, 2, 3, 4, ...

When counting individual objects, one is the smallest number, and it is not necessary, and sometimes impossible, to divide it into shares, however, even with rough measurements of quantities, one has to divide 1 into shares. Historically, the first extension of the concept of number is the addition of fractional numbers to a natural number. The introduction of fractional numbers is associated with the need to make measurements. The measurement of any value consists in comparing it with another, qualitatively homogeneous with it and taken as a unit of measurement. This comparison is carried out by means of a method-specific operation of "setting aside" the unit of measure on the measurand and counting the number of such setbacks. This is how the length is measured by setting aside a segment taken as a unit of measurement, the amount of liquid is measured using a measuring vessel, etc.

A fraction is a part (share) of a unit or several equal parts of it.

Designated: where m and n are integers; - fraction reduction; - extension. Fractions with a denominator of 10 n, where n is an integer, are called decimal.

Among decimal fractions special place occupy periodic fractions: - pure periodic fraction, - mixed periodic fraction

Further expansion of the concept of number is already caused by the development of mathematics itself (algebra). Descartes in the 17th century introduces the concept negative number, which gave its geometric interpretation as the direction of the segments. Creation by Descartes analytical geometry, which made it possible to consider the roots of the equation as the coordinates of the points of intersection of some curve with the abscissa axis, finally erased the fundamental difference between the positive and negative roots of the equation, their interpretation turned out to be essentially the same.

Numbers whole (positive and negative), fractional (positive and negative) and zero are called rational numbers. Any rational number can be written as a finite and periodic fraction.

The set of rational numbers turned out to be insufficient for studying continuously changing variables. Here, a new extension of the concept of number turned out to be necessary, consisting in the transition from the set of rational numbers to the set of real (real) numbers. The introduction of real numbers occurred by adding irrational numbers to rational numbers: irrational numbers are endless decimal non-periodic fractions.

Irrational numbers appeared when measuring incommensurable segments (the side and diagonal of a square), in algebra - when extracting roots, an example of a transcendental, irrational number is π, e.

A clear definition of the concept of a real number is given by I. Newton, one of the founders of mathematical analysis, in "General Arithmetic": "By number, we mean not so much a set of units, but an abstract ratio of some quantity to another quantity of the same kind, which we take as a unit." This formulation gives a single definition of a real number, rational or irrational. Later, in the 70s. 19th century, the concept of a real number was refined on the basis of a deep analysis of the concept of continuity in the works of R. Dedekind, G. Kantor and K. Weierstrass.

According to Dedekind, the continuity property of a straight line is that if all the points that make up a straight line are divided into two classes so that each point of the first class lies to the left of each point of the second class (“break” the straight line into two parts), then either in the first class there is the rightmost point, or in the second - the leftmost point, i.e., the point at which the “break” of the line occurred.

The set of all rational numbers does not have the continuity property. If the set of all rational numbers is divided into two classes so that each number of the first class is less than each number of the second class, then with such a partition (Dedekind's "section") it may turn out that in the first class there will be no largest number, and in the second - least. So it will be, for example, if all negative rational numbers, zero and all positive numbers whose square is less than two, are assigned to the first class, and all positive numbers whose square is greater than two are assigned to the second. Such a cut is called irrational. Then the following definition of an irrational number is given: each irrational section in the set of rational numbers is associated with an irrational number, which is considered greater than any number of the first class, and less than any number of the upper class. The totality of all real numbers, rational and irrational, already has the property of continuity.

Cantor's rationale for the concept of a real number differs from Dedekind's, but is also based on an analysis of the concept of continuity. Both Dedekind's definition and Cantor's definition use the abstraction of actual infinity. Thus, in Dedekind's theory, an irrational number is determined by means of a section in the totality of all rational numbers, which is conceived as given as a whole.

All real numbers can be shown on the number line. Numerical axis (number line):

a) a horizontal straight line with a direction chosen on it;

b) reference point - point 0;

c) scale unit

[Great Soviet Encyclopedia. - [Electronic resource]. URL: http://dic.academic.ru/dic.nsf/bse/150404/Number].

To date, there are seven generally accepted levels of generalization of numbers: natural, rational, real, complex, vector, matrix and transfinite numbers. Some scientists propose to consider functions as functional numbers and expand the degree of generalization of numbers to twelve levels.

[Anishchenko Evgeny Alexandrovich. "Number as a Basic Concept of Mathematics". - [Electronic resource]. URL: http://www.referat.ru/referats/view/7401].

Russian scientist Ozolin E.E. expressed an important thought that very accurately conveys the modern intellectual atmosphere in the mathematical community. Everyone knows that number theory is the most complex and important branch of mathematics. Nevertheless, the theory of numbers seems to be overlooked. Whereas the most insignificant changes in this theory can cause a "storm" in all sections of mathematics [Ozolin E.E. (Ozes) October 2004. The concept of a number. - [Electronic resource]. URL: http://ozes-world.narod.ru/MtMetaMt/1_4/Mt1_4.htm].

Moreover, - E.E. Ozolin writes with surprise, - despite the fact that the ancient Greeks knew far from everything about numbers, the sadder is the fact that “modern mathematicians (not to mention all the others) have concepts and knowledge the number is sometimes inferior to ancient Greek.

This, you see, is already nonsense” [Ibid.].

As a confirmation of this consideration, E.E. Ozolin conducts a historical analysis of the principles for constructing the concept of number and comes to the following conclusion. European mathematics, especially since the 13th century, builds the concept of number according to the principle of nesting of Thales spheres, “that is, the set of natural numbers is invested in the set of integers, the set of integers is invested in the set of rational numbers, the set of rational numbers is invested in the set of real numbers, the set of real numbers numbers are embedded in the set of complex numbers, etc.)” [Ibid.]. “And, despite the fact that both Kurt Gödel from the standpoint of formal logic (back in 1931), and myself from the standpoint of metamathematics, have long proved and re-proved that the five-layer structure of nested spheres cannot be complete and logically correct, we again and again we are faced with erroneous "school dogmas" in the form of supposedly fair statements that, for example, natural numbers are a subset of rational numbers.

Therefore, once again I want to draw your attention to the fact that this cannot be. For example, in the framework of mathematics, we can only talk about the formal equality of the natural number 1 (one) to the rational number 1.00(0) to one. At the same time, the logical, mathematical (and physical!) meaning of these numbers is completely different. For example, a natural unit is a number that, when added to the existing one, gives the next number, a rational unit is a number, when multiplied by which given number does not change the meaning! How can a unit change the meaning” [Ibid.] ???

“Moreover, - continues E.E. Ozolin, - natural and rational numbers belong to completely different metalogical structures. Therefore, we cannot even talk about the formal mathematical relationship of these numbers.

At first glance, it may seem that the problem of the logical difference between natural and rational numbers that I have indicated is “not worth a damn”. And the majority of mathematicians, even if they agree with me, will surely say that “units are also units in Africa, and what difference does it make what mathematical and logical meaning to put into them, the same or different” [Ibid.].

But such a view is a big misconception - a "harmful myth of school education", which has no mathematical and logical basis. “And upon closer and more detailed consideration, it turns out that the difference in the logical sense of natural and rational numbers entails quite serious consequences. practical application mathematics" [ibid.].

And in conclusion, E.E. Ozolin makes the following frank conclusion: “ ... mathematics is a very free science, and the rigor of mathematics is only apparent. In mathematics, you can build any, the most incredible axiomatic structures, and explore them, no matter how meaningless and abstract from reality they are. In other words, invent and try to your heart's content. In metamathematics, it is practically impossible to do this, and all the structures of metamathematics, one way or another, are connected with reality. Paradoxical as it may seem, the reality turns out to be much richer than "our imagination".“[Ibid.].

Returning to the immediate topic of our study, we can draw the following conclusion. All types of numbers (numerical sets) have a different logical nature and mathematical properties. Therefore, it is methodologically incorrect to build the theory of numbers by direct generalization. The paramount relation of this conclusion concerns the natural and real numbers. The unit of natural numbers and the unit of real numbers have completely different origins and different mathematical properties. You cannot measure the number of apples with a ruler; equally, it is impossible, only knowing how to count and not having a ruler at hand, to measure the length of the table. One cannot be reduced to the other. The unit of the former is indivisible, while the unit of the latter is necessarily divisible. The natural integers are actually numbers in the strict sense, while the real numbers belong to such a form of quantity as magnitude. The confusion of the form of number and magnitude, which goes back to the Pythagoreans, is the main source of the modern crisis in mathematics and the most important prerequisite for the arithmetization of geometry and G. Kantor's set theory, since the idea of ​​constructing divisible mathematical objects from indivisible ones underlies the construction of G. Kantor's concept of actual infinity.

One more note. Modern mathematicians not only do not understand the nature of a number, as E. Ozolin correctly noted, but also do not understand the logical and mathematical nature of a quantity and other fundamental concepts of mathematics (for example, a set).

Here, for example, is what famous mathematicians write about the value:

“Value is one of the basic mathematical concepts, the meaning of which, with the development of mathematics, was subjected to a number of generalizations,” writes A.N. Kolmogorov [Kolmogorov A.N. Value. - TSB. - T. 7. - M., 1951. C. 340]. “This ... theory - the doctrine of magnitude - hardly plays essential role in the matter of substantiating the whole of mathematics,” wrote the prominent Soviet mathematician V.F. Kagan [Kagan V.F. Essays on Geometry. - M.: Moscow University, 1963. S. 109].

Let us dwell on the latter, in which the meaning of the concept of quantity is the most consistent and clear. “... for a mathematician,” wrote V.F. Kagan, “the value is completely determined when the set of elements and comparison criteria are indicated” [Ibid., P. 107]. In other words, the value is a set of homogeneous objects, the comparison of the elements of which allows us to use the terms "equal", "greater", "less". A counter question arises, if we compare a certain set of natural numbers with another certain set of the same natural numbers, for example, the numbers 5 and the numbers 7, can we apply the above terms to them? The question is rhetorical. The proposed definition of the concept of magnitude, in fact, indicates that its author does not distinguish between these two at all. fundamental concepts(numbers and magnitudes). Proponents of set theory and Cantor himself also lamented that the basic concept of this theory is also difficult to define. E. Ozolin in his article notes that it is very difficult to define mathematics as a subject [Ozolin E.E. (Ozes) October 2004. The concept of a number. - [Electronic resource]. URL: http://ozes-world.narod.ru/MtMetaMt/1_4/Mt1_4.htm].

To make sure that all these doubts are unfounded, it is necessary to return again to Aristotle, who, in several definitions, gives exhaustive answers to our questions.

“Quantity is that which is divisible into component parts, each of which, whether there are two or more of them, is by nature something one and a definite something. Every quantity is a set if it is countable, and magnitude is if it is measurable. A set is what is divisible into non-continuous parts, a quantity - into continuous parts ... Of all these quantities, a limited set is a number, a limited length is a line, a limited width is a plane, a limited depth is a body ”[Aristotle. Op. in four volumes. T.1. Metaphysics. P.164].

From this fragment of Aristotle. We obtain the following rigorous definitions.

Mathematics is a science whose subject is pure quantity.

Quantity is that which is divisible into its component parts, each of which, whether there are two or more of them, is by nature something one and a definite something.

A set is a quantity that is countable, i.e. divisible into non-continuous parts.

A quantity is a quantity that is measurable, i.e. divide into parts continuous

Number is a limited set.

The line is limited in length.

The plane is a limited width.

The body is limited depth.

From these provisions follows:

The unit of a number has no dimension, it is a unit of account, i.e. indivisible, since we only count in whole numbers.

The unit of magnitude is always divisible.

The unit of number is the purest form of abstract quantity, i.e. it is a form indifferent to geometric space.

The unit of magnitude is the pure quantity plus the geometric space.

Geometric space is an abstraction of physical reality. Physical reality has a qualitative certainty and extension. If we abstract from the qualitative certainty of physical reality, we get a geometric space.

Formally, both the unit of a number and the unit of magnitude are a number, but the essence and mathematical properties of these numbers are different. From the unit of a number it is impossible to obtain a unit of magnitude. Whereas from the value you can get a pure number. To do this, it is necessary to abstract from the geometric space - the dimension. These points are well analyzed in Aristotle's Physics.

Therefore, from a number (in the strict sense) it is impossible to obtain a value. And since the subject of arithmetic is the concept of number, and the subject of geometry is magnitude, then geometry cannot be reduced to arithmetic. These are different ways of existence of the quantitative certainty of the material world.

Thus, at the heart of modern mathematics lies a deep delusion - the illegal identification of number and magnitude, arithmetic and geometry. The concept of magnitude is more fundamental, because from it we can derive the concept of number. In addition, this concept "connects" mathematics with physics, creates obstacles for unjustified formalization and speculative constructions. Therefore, the arithmetization of geometry led to the degeneration of the subject of mathematics, its formalization (Bourbakization) and the theory of transfinite numbers. The arithmetization of mathematics is, in fact, the process of reducing the subject of mathematics to a number.