One and the same object can have many models, and different objects can be described by one model. Classification of knowledge representation models

Mathematical analysis is a branch of mathematics that deals with the study of functions based on the idea of an infinitely small function.

The basic concepts of mathematical analysis are quantity, set, function, infinitesimal function, limit, derivative, integral.

Value everything that can be measured and expressed by a number is called.

many is a collection of some elements united by some common feature. The elements of a set can be numbers, figures, objects, concepts, etc.

Sets are denoted by capital letters, and elements of a set by lowercase letters. Set elements are enclosed in curly braces.

If element x belongs to the set X, then write x∈X (∈ - belongs).
If set A is part of set B, then write A ⊂ B (⊂ - is contained).

A set can be defined in one of two ways: by enumeration and by a defining property.

For example, the enumeration defines the following sets:

A=(1,2,3,5,7) - set of numbers
Х=(x 1 ,x 2 ,...,x n ) is a set of some elements x 1 ,x 2 ,...,x n
N=(1,2,...,n) is the set of natural numbers
Z=(0,±1,±2,...,±n) is the set of integers

The set (-∞;+∞) is called number line, and any number is a point of this line. Let a be an arbitrary point on the real line and δ a positive number. The interval (a-δ; a+δ) is called δ-neighbourhood of the point a.

The set X is bounded from above (from below) if there is such a number c that for any x ∈ X the inequality x≤с (x≥c) is satisfied. The number c in this case is called top (bottom) edge sets X. A set bounded both above and below is called limited. The smallest (largest) of the upper (lower) faces of the set is called exact top (bottom) face this set.

Basic Numeric Sets

N	(1,2,3,...,n) The set of all
Z	(0, ±1, ±2, ±3,...) Set whole numbers. The set of integers includes the set of natural numbers.
Q	A bunch of rational numbers. In addition to whole numbers, there are also fractions. A fraction is an expression of the form , where p is an integer, q- natural. Decimals can also be written as . For example: 0.25 = 25/100 = 1/4. Integers can also be written as . For example, in the form of a fraction with a denominator of "one": 2 = 2/1. Thus, any rational number can be written as a decimal fraction - finitely or infinitely periodic.
R	Many of all real numbers. Irrational numbers are infinite non-periodic fractions. These include: Together, two sets (rational and irrational numbers) form the set of real (or real) numbers.

If a set contains no elements, then it is called empty set and recorded Ø .

Elements of logical symbolism

The notation ∀x: |x|<2 → x 2 < 4 означает: для каждого x такого, что |x|<2, выполняется неравенство x 2 < 4.

quantifier

When writing mathematical expressions, quantifiers are often used.

quantifier is called a logical symbol that characterizes the elements following it in quantitative terms.

∀- general quantifier, is used instead of the words "for all", "for anyone".
∃- existential quantifier, is used instead of the words "exists", "has". The symbol combination ∃! is also used, which is read as there is only one.

Operations on sets

Two sets A and B are equal(A=B) if they consist of the same elements.
For example, if A=(1,2,3,4), B=(3,1,4,2) then A=B.

Union (sum) sets A and B is called the set A ∪ B, whose elements belong to at least one of these sets.
For example, if A=(1,2,4), B=(3,4,5,6), then A ∪ B = (1,2,3,4,5,6)

Intersection (product) sets A and B is called the set A ∩ B, whose elements belong to both the set A and the set B.
For example, if A=(1,2,4), B=(3,4,5,2), then A ∩ B = (2,4)

difference sets A and B is called a set AB, the elements of which belong to the set A, but do not belong to the set B.
For example, if A=(1,2,3,4), B=(3,4,5), then AB = (1,2)

Symmetric difference sets A and B is called the set A Δ B, which is the union of the differences of the sets AB and BA, that is, A Δ B = (AB) ∪ (BA).
For example, if A=(1,2,3,4), B=(3,4,5,6), then A Δ B = (1,2) ∪ (5,6) = (1,2,5 .6)

Properties of set operations

Permutability properties

A ∪ B = B ∪ A
A ∩ B = B ∩ A

associative property

(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)

Countable and uncountable sets

In order to compare any two sets A and B, a correspondence is established between their elements.

If this correspondence is one-to-one, then the sets are called equivalent or equivalent, A B or B A.

Example 1

The set of points of the leg BC and the hypotenuse AC of the triangle ABC are of equal power.

Mathematical set

A bunch of- one of the key objects of mathematics, in particular, set theory. “Under the set we mean the unification into one whole of certain, completely distinguishable objects of our intuition or our thought” (G. Kantor). This is not in the full sense a logical definition of the concept of a set, but only an explanation (because to define a concept means to find such a generic concept in which this concept is included as a species, but a set is perhaps the broadest concept of mathematics and logic).

theories

There are two main approaches to the concept of a set - naive and axiomatic set theory.

Axiomatic set theory

Today, a set is defined as a model that satisfies the ZFC axioms (the Zermelo-Fraenkel axioms with the axiom of choice). With this approach, in some mathematical theories, collections of objects arise that are not sets. Such collections are called classes (of different orders).

Set element

The objects that make up a set are called set elements or set points. Sets are most often denoted by capital letters of the Latin alphabet, its elements - by small ones. If a is an element of the set A, then write a ∈ A (a belongs to A). If a is not an element of the set A, then write a ∉ A (a does not belong to A).

Some kinds of sets

An ordered set is a set on which the order relation is given.
A set (in particular, an ordered pair). Unlike just a set, it is written inside parentheses: ( x 1 , x 2 , x 3 , …), and elements can be repeated.

By hierarchy:

Set of sets Subset Superset

By limitation:

Operations on sets

Literature

Stoll R.R. Sets. Logics. axiomatic theories. - M .: Education, 1968. - 232 p.

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. Kantor in his work Beiträge zur Begründung der transfiniten Mengenlehre gives a definition: by "set" we mean the combination into a certain whole of certain well-distinguishable objects of our contemplation or our thinking (which will be called "elements" of the set). 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) - (from other Greek παράδοξος - unexpected, strange from other Greek παρα-δοκέω - I seem) - 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 a 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 of formal logic and the 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 a new branch of mathematics and thereby 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 is an arbitrary set of ordinal numbers, then the sum-set is an ordinal number greater than or equal to each of the elements of . 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 ordinal type is formed, 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 errors of the latter 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 can contain elements defined only in terms of a set, as well as elements that presuppose 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.

Cantor's paradox. Assume that the set of all sets exists. In this case, it is true that every set is a subset of . But it follows from this that the cardinality of any set does not exceed the cardinality of . But by virtue of the axiom of the set of all subsets, for , as well as any set, there is a set of all subsets , and by Cantor's theorem, which contradicts the previous statement. Therefore, it cannot exist, which conflicts with the "naive" hypothesis that any syntactically correct logical condition defines a set, i.e. that for any formula not containing free. 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 the predicate retain their volume and content, in this case there is a transition from one category to another, which results in a discrepancy;
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, in order to define itself, would require the definition of a set;
basic logical laws are violated:
1. the law of identity is violated when the non-identity of the subject and the predicate is revealed;
2. the law of contradiction - when two contradictory judgments are derived with the same right;
3. 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.

Russell's paradox. Here is one of his options. Let be the set of all sets that do not contain themselves as their element. Does it contain itself as an element? If so, then, by definition, it shouldn't be an element - a contradiction. If not - then, by definition, it must be an element - 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 finds that it took him a whole year to recount the events of the first day of his life, and another year 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 -th day for the -th 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. only the “axiom of common sense” leads to a contradiction. 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 of the infinite life of the protagonist, there are two infinite sets of equal power, which, if we consider 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 said to be equally composed if one can be divided into a finite number of parts, moved, and made up of the second. More precisely, two sets and are equally composed if they can be represented as a finite union of disjoint subsets and such that for each 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 of the Euclidean plane that has two disjoint subsets, each of which can be divided into a finite number of parts, so that their can be translated by isometries into a covering of the set . 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: Required to name "the 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.
Grelling-Nilson paradox: words or signs can denote some 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 paradox: using Godel's completeness theorem and the Löwenheim-Skolem theorem, we obtain that the axiomatic set theory remains true even when only a countable set of sets is assumed (available) for its interpretation. At the same time, the axiomatic theory includes the already mentioned Cantor's theorem, which leads 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 the concepts of 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 an 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.

A more or less complete idea of finite methods is 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 a 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 any 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 cannot cover the entire realm 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 (the ideas of the atomists, the aporias of Zeno of Elea, the infinitesimal methods in antiquity, the calculus of infinitesimals 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 one 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 function theory Baire, Lebesgue and Borel was directed against the acceptance of the axiom of free choice, which proves Zermelo's theorem, which states 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. Heiting 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 of the general properties of objects) as a way of constructing and constructively understanding abstract concepts, mathematical judgments; proof of the consistency of consistent theories. In the formal aspect, the application of the 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.

Thus, the epistemological aspect of mathematical knowledge allows us to evaluate the revolutionary changes at the stage of the crisis of the foundations of mathematics at the turn of the 19th-20th centuries. from new positions in understanding the process of cognition, the nature and role of the subject in it. The epistemological subject of the traditional theory of knowledge, corresponding to the period of domination of the set-theoretical 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 copying 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 emergence of a rigorous concept of an algorithm in the 1930s, 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. Whether such an approach is definitive is debatable, 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

Georg Cantor. Beiträge zur begründung der transfiniten mengenlehre. Mathematische Annalen, 46:481-512, 1895.
I.N. Burova. Paradoxes of set theory and dialectics. Science, 1976.
M.D. Potter. Set theory and its philosophy: a critical introduction. Oxford University Press, Incorporated, 2004.
Zhukov N.I. Philosophical foundations of mathematics. Minsk: Universitetskoe, 1990.
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.
O. M. Mizhevich. Two Ways to Overcome Paradoxes in G. Kantor's Set Theory. Logical and Philosophical Studies, (3):279-299, 2005.
S. I. Masalova. PHILOSOPHY OF INTUITIONIST MATHEMATICS. Bulletin of DSTU, (4), 2006.
Chechulin V.L. Theory of sets with self-ownership (foundations and some applications). Perm. state un-t. – Perm, 2012.
S. N. Tronin. Brief summary of lectures on the discipline "Philosophy of Mathematics". Kazan, 2012.
Grishin V.N., Bochvar D.A. Studies in set theory and non-classical logics. Science, 1976.
Hofstadter D. Gödel, Escher, Bach: this endless garland. Bahrakh-M, 2001.
Kabakov F.A., Mendelson E. Introduction to mathematical logic. Publishing house "Nauka", 1976.
YES. Bochvar. On the question of the paradoxes of mathematical logic and set theory. Mathematical Collection, 57(3):369-384, 1944.

The description of the subject area (the creation of its ontology) begins with the selection of objects and their classification, which traditionally consists in compiling a tree of subclasses and assigning individuals to them. At the same time, the term "class", in fact, is used in the meaning of "set": referring an object to a class is thought of as including it as an element in the corresponding set. The purpose of this text is to show that such a unified approach to describing the structure of the subject area is a strong simplification and does not allow fixing the variety of semantic relations of objects.

Let's look at three options for classifying the Bug individual:

Animal - dog - husky - Bug.
Service - riding - Bug.
Kennel - team of dogs - Zhuchka.

The first sequence of subordinate entities is unambiguously described by specifying classes and subclasses: the bug is an individual of the “like” class, the “like” class is a subclass of dogs, and that one is a subclass of the “animal” class. In this case, the class "animals" is treated as a set of all animals, and the class "likes" as a subset of the set "dogs". However, such a description, despite the fact that it is quite clear, is meaningfully tautological, self-referential: we call the individual Bug a husky if it is included in the set of huskies, and the set of huskies itself is defined as the totality of all individuals of the huskies - that is, inclusion in the set of meaningful duplicate name. In addition, the description of a class-set is completely exhausted by the description of an individual falling under the concept defining the class. It should also be noted that the operation of such classes-sets does not depend on the number of elements in them: the Bug's husky will be a husky even when it remains the only, last husky on Earth. Moreover, we can operate with such classes-sets even in the absence of individuals in them: we can build an ontology of already extinct dinosaurs, think of a class that only in the future will include a unique device being designed, or build a model of the subject area of mythical animals, heroes of fairy tales, although at the same time the cardinality of all class-sets will be equal to zero.

So, if we talk about the content side of the analyzed classification (animal - dog - husky - Bug), then it (the content side) cannot be expressed in any way through the relation of sets and subsets. In this case, we are dealing with conceptualization - the selection of concepts and establishing genus-species relations between them. At the same time, the actual number of elements of the conceptual class, that is, the scope of the concept, does not appear in its definition and is mentioned (and even then not meaningfully) only when one concept (“like”) falls under another (“dog”), that is, when as a kind of genus. Yes, we can state that the scope of the concept "dog" is greater than the scope of the concept "like", but the real numerical ratio of these sets does not have any ontological meaning. Exceeding the volume of a class of the volume of a subclass in genus-species relations only reflects the fact that, according to the definition of a genus, it should include several species - otherwise this classification becomes meaningless. That is, in the genus-species conceptual classification, we are interested in the content of concepts - how the type "dog" differs from the type "cat" (which also falls under the generic concept of "animal" for them), and not how the volumes of the sets of the genus and species are related and even more so the volumes of specific concepts (“dog” and “cat”). And in order to distinguish conceptual classes from truly countable sets, it would be more correct to speak of falling under the concept and not about inclusion it into a class/set. It is clear that in the formal notation, the statements “belongs to the concept of X” and “is an element of the class X” may look the same, but not understanding the essential difference between these two descriptions can lead to serious errors in the construction of the ontology.

In the second variant (service - driving - Bug), we are also not interested in comparing the concept of "driving" with any set: the semantic content of the statement "Bug - driving" does not depend on whether it is the only driving one or there are many of them. It would seem that here we are dealing with genus-species relations: the concept of "driving" can be considered as a species relative to the generic concept of "service". But the connection of the individual "Bug" with the concept of "driving" differs significantly from the connection with the concept of "like": the second, conceptual, concept is immanent and invariably inherent in the individual, and the first reflects the local in time specialization. The bug was not born as a rider, and perhaps with age it may cease to be it and move into the category of guards, and in old age, in general, lose any “profession”. That is, speaking of specialization, we can always single out the events of acquiring and losing connection with a particular concept. For example, the Bug could be recognized as the absolute champion of the breed, and then lose this title, which is fundamentally impossible with conceptual concepts: the Bug from birth to death, that is, for the entire time period of its existence as an individual, is a dog and a husky. So a person remains the concept of “man” all his life, but situationally (from event to event) can fall under the specializing concepts of “schoolchild”, “student”, “doctor”, “husband”, etc. And as already noted, the connection with these concepts does not in the least mean inclusion in a certain set (although it may look like this) - the assignment of a specializing concept is always the result of a specific relationship of an individual with other individuals: entering a school, university, obtaining a diploma, registering a marriage, etc. Therefore, specializing concepts can also be called relational. From the above examples, another significant difference between the conceptual classification and specialization follows: an individual can have several specializations (a bug can be a sled dog and a champion of the breed, a person is a student and a husband), but cannot simultaneously enter more than one conceptual hierarchy (a bug cannot be a dog , and a cat).

And only in the third version of the description of Zhuchka - as belonging to a certain kennel and as a member of a specific team pulling sleds across the tundra - it is simply necessary to mention the multitude. Only in this case, we have the right to say that an individual is an element of a concrete set with a countable number of elements, and does not fall under the concept, which can be represented as an abstract set, conditionally fixing the scope of this concept. And here it is important that an individual is a part of another individual, initially defined as a set: a kennel and a team are necessarily a non-empty set of dogs, and the number of elements of this set is necessarily included in their definitions as individuals. That is, in this case, we should talk about the relationship part-whole: The bug is part of the kennel and part of the team. Moreover, the entry or non-entry of the Bug into a particular team changes its (team) content: if we had a team-two, then after the removal of the Bug, the team turns into a single team. In such cases, we are dealing not just with a countable set (dogs in a kennel), but with an individual whose essence changes when the composition of its elements changes, is determined by this composition, that is, with system. If a kennel is just an individual-group, described through a set of elements included in it, then a team is a system, the essence of which depends on the number and specifics of its parts.

Consequently, when constructing an ontology of a subject area, one can single out real objects-sets, defined precisely as a collection of a certain number of individuals. These are: a class at school, goods in a box in a warehouse, parts of an electronic device block, etc. And these sets can be subsets of other real countable sets: all students in a school, all goods in a warehouse, all parts of a device. When distinguishing these sets, it is essential that they (these sets) act as independent individuals (a team, a batch of goods, a set of parts), the main attribute of which is precisely the number of elements included in them. Moreover, a change in this attribute can lead to a change in the status of the object, for example, with an increase in the number of elements, turn a quartet into a quintet or a regiment into a brigade. It is also important that the description of these set-objects, complex objects, is not limited to the description of the individuals included in them, although it may include an indication of the admissible type of the latter (a string quartet, a team of horses). And such relationships - not between abstract sets, but between sets that are individuals, complex objects - are more accurately described as part-whole relationships, and not class-subclass.

So, the traditional classification of individuals by assigning them to certain classes-sets cannot be considered homogeneous. It is necessary to distinguish between (1) the inclusion of individuals as parts in a complex object (whole), the semantic specificity of which is not limited to the description of its elements. At the same time (1.1.), an object-whole can be considered only as a named set of individuals (parts in a package, a collection of paintings), for which, in fact, only the number of parts is important. Such objects can be called groups (or collections)). Also (1.2.) an object-whole can be meaningfully (and not just quantitatively) determined by its parts and, as a result, have attributes that parts do not have. Such integrity is traditionally called systems, and parts of systems - elements. The second option for describing objects by assigning them to subclasses is (2) the falling of individuals under the concept, which can only be formally, tautologically described as the inclusion of individuals in a set whose power is equal to the power of the concept. The conceptual description of individuals, in turn, can be classified into (2.1) conceptual, globally fixing the type of the individual, and (2.2) specialized (relational), locally in time and space (event-wise) connecting the individual with other objects.

The above reasoning, first of all, raises the question of the sufficiency and adequacy of the traditional approach to describing the subject area using a classification based on set theory. And the conclusion is proposed: in order to fix the whole variety of object relationships in ontologies, more differentiated classification tools (groups, systems, conceptual and specializing concepts) are needed. The formalism of set theory can only be used as a local simplification for the needs of inference, and not as the main method of description.

Portal for the student. Self-training

One and the same object can have many models, and different objects can be described by one model. Classification of knowledge representation models

Basic Numeric Sets

Elements of logical symbolism

Operations on sets

Properties of set operations

Countable and uncountable sets

theories

Axiomatic set theory

Set element

Some kinds of sets

Operations on sets

Literature

see also

See what "Mathematical set" is in other dictionaries:

Books

Portal for the student. Self-training

Basic Numeric Sets

Elements of logical symbolism

Operations on sets

Properties of set operations

Countable and uncountable sets

theories

Axiomatic set theory

Set element

Some kinds of sets

Operations on sets

Literature

see also

See what "Mathematical set" is in other dictionaries:

Books

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