Games of the mathematical mind. Theocentric analysis of G. Kantor's set theory

Lecture 12: Basic concepts of set theory

Considering a system as a set of elements makes it possible to use the apparatus of set theory for its mathematical description. Moreover, in a number of important cases, the connections between elements are conveniently described using the apparatus of mathematical logic.

The concept of a set is one of those fundamental concepts of mathematics that are difficult to give precise definition using elementary concepts. Therefore, we confine ourselves to a descriptive explanation of the concept of a set.

many called a set of certain quite distinct objects, considered as a single whole. The creator of set theory, Georg Cantor, gave the following definition of a set - "a set is a lot that we think of as a whole."

The individual objects that make up a set are called elements sets.

Sets are usually denoted by capital letters of the Latin alphabet, and the elements of these sets are denoted by small letters of the Latin alphabet. Sets are written in curly brackets ( ).

It is customary to use the following notation:

• a ∈ X — “the element a belongs to the set X”;
• a ∉ X — “element a does not belong to the set X”;
• ∀ - quantifier of arbitrariness, generality, denoting "any", "whatever", "for all";
• ∃ is an existence quantifier: ∃y ∈ B — “there is (there is) an element y from the set B”;
• ∃! —existence and uniqueness quantifier: ∃!b ∈ C — “there is a unique element b from the set C”;
• : - “such that; possessing the property";
• → - a symbol of the consequence, means "entails";
• ⇔ - quantifier of equivalence, equivalence - "if and only then".

Sets are final and endless. The sets are called final, if the number of its elements is finite, i.e. if there is a natural number n, which is the number of elements of the set. A \u003d (a 1, a 2, a 3, ..., a n). The set is called endless if it contains an infinite number of elements. B=(b 1 ,b 2 ,b 3 , ...). For example, the set of letters of the Russian alphabet is a finite set. The set of natural numbers is an infinite set.

The number of elements in a finite set M is called the cardinality of the set M and is denoted by |M|. empty set - a set that does not contain a single element - ∅. The two sets are called equal, if they consist of the same elements, i.e. are the same set. Sets are not equal to X ≠ Y if X has elements that do not belong to Y, or Y has elements that do not belong to X. The set equality symbol has the following properties:

• X=X; - reflexivity
• if X=Y, Y=X - symmetry
• if X=Y,Y=Z, then X=Z is transitive.

According to this definition of equality of sets, we naturally obtain that all empty sets are equal to each other, or that it is the same that there is only one empty set.

Subsets. Inclusion relation.

The set X is a subset of the set Y if any element of the set X ∈ and the set Y. Denoted by X⊆Y.

If it is necessary to emphasize that Y contains other elements besides elements from X, then the strict inclusion symbol ⊂: X⊂Y is used. The relationship between the symbols ⊂ and ⊆ is given by:

X⊂Y ⇔ X⊆Y and X≠Y

We note some properties of the subset that follow from the definition:

1. X⊆X (reflexivity);
2. → X⊆Z (transitivity);
3. ∅ ⊆ M. It is customary to assume that the empty set is a subset of any set.

The original set A in relation to its subsets is called complete set and is denoted by I.

Any subset A i of a set A is called a proper set of A.

A set consisting of all subsets given set X and an empty set ∅ is called boolean X and is denoted by β(X). Boolean power |β(X)|=2 n .

Countable set- this is such a set A, all elements of which can be numbered in a sequence (perhaps infinite) a 1, a 2, a 3, ..., a n, ... so that each element receives only one the number n and each natural number n would be given as a number to one and only one element of our set.

A set equivalent to the set of natural numbers is called a countable set.

Example. The set of squares of integers 1, 4, 9, ..., n 2 is only a subset of the set of natural numbers N. The set is countable, since it is brought into one-to-one correspondence with the natural series by assigning to each element the number of that number of the natural series, the square which he is.

There are 2 main ways to define sets.

• enumeration (X=(a,b), Y=(1), Z=(1,2,...,8), M=(m 1 ,m 2 ,m 3 ,..,m n ));
• description - indicates the characteristic properties that all elements of the set have.

A set is completely defined by its elements.

An enumeration can only specify finite sets (for example, a set of months in a year). Infinite sets can only be defined by describing the properties of its elements (for example, the set of rational numbers can be defined by describing Q=(n/m, m, n∈Z, m≠0).

Ways to specify a set by description:

a) by specifying a generating procedure with an indication of the set (sets) that the parameter (parameters) of this procedure runs through - recursive, inductive.

X=(x: x 1 =1, x 2 =1, x k+2 =x k +x k+1 , k=1,2,3,...) - many Fibonicci numbers.

(multiple elements x, such that x 1 =1, x 2 =1 and arbitrary x k+1 (for k=1,2,3,...) is calculated by the formula x k+2 = x k + x k+1 ) or X= Extensions

Main article: Kit theory

Set theory is a natural extension (generalization) of set theory. Like a set, a set is a set of elements from a certain area. Difference from the set: kits allow the presence several instances of the same element (the element is included from zero times, that is, not included, up to any given number of times). (see, for example, Multicombinations).

Applications

see also

Notes

Literature

• K. Kuratovsky , A. Mostovsky Set theory / Translated from English by M. I. Briefly, edited by A. D. Taimanov. - M .: Mir, 1970. - 416 p.
• N. K. Vereshchagin, A. Shen. Lectures on mathematical logic and the theory of algorithms. Part 1. Beginnings of set theory.
• A. Frenkel, I. Bar-Hillel Foundations of set theory / Translation from English by Yu. A. Gastev, edited by A. S. Yesenin-Volpin. - M .: Mir, 1966. - 556 p.

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• Mathematical analysis
• Subset

See what "Set Theory" is in other dictionaries:

SET THEORY- SET THEORY, a branch of mathematics that began with the work of George Boole in the field of mathematical logic, but is currently more associated with the study of SETs of abstract or real objects, and not with logical ones ... ... Scientific and technical encyclopedic dictionary

set theory- — [Ya.N. Luginsky, M.S. Fezi Zhilinskaya, Yu.S. Kabirov. English-Russian Dictionary of Electrical Engineering and Power Industry, Moscow, 1999] Topics in electrical engineering, basic concepts of EN set theory ... Technical Translator's Handbook

SET THEORY- a theory in which sets (classes) of elements of an arbitrary nature are studied. Created primarily by the works of Cantor (as well as R. Dedekind and K. Weierstrass), T. m. by the end of the 19th century. became the basis for the construction of the mathematical methods that had developed by that time. ... ... Philosophical Encyclopedia

SET THEORY- branch of mathematics that studies general properties sets. A set is any union into one whole of some definite and different objects of our perception or thought. In T. m., the general properties of various operations are studied ... ... encyclopedic Dictionary in psychology and pedagogy

Georg Ferdinand Ludwig Philipp Kantor(in my opinion and, I think, not only in my opinion) - one of the greatest mathematicians in the history of mankind. It's pathetic, maybe too much, but it's sincere))

It was he who founded the theory of sets (perhaps a little in the form in which we know it now).
It's hard to believe, but he was the first to introduce the concept in mathematics sets and gave it an informal definition. And it happened in the second half of the XIX century.
Previously, sets were not operated on in mathematics!
The set theory put forward by Cantor was later called Naive set theory.

concept sets is now included among the so-called primary, undefined concepts. Such as, suppose dot in mathematics or information in information theory.
Cantor himself defined a set as follows: "many is many thought as one".

Kantor developed a program for the standardization of mathematics, which was based precisely on the concept sets. Any mathematical object had to be considered as a "set".
For example, the natural series is a set that satisfies Peano's axioms. Each natural number taken separately is also a set, but consisting of only one element.

The term "set theory" itself was introduced into mathematics later. Kantor called his theory "Mengenlehre" - the doctrine of sets.

The appearance of the Mengenlehre caused a serious battle in mathematical circles. The doctrine had both ardent admirers (among the outstanding mathematicians of that time) and ardent opponents.

But in its original form, the theory was not viable.

Here is what Wikipedia says:
However, it soon became clear that Cantor's attitude to unlimited arbitrariness when operating with sets (expressed by himself in the principle "the essence of mathematics lies in its freedom") is inherently vicious. Namely, a number of set-theoretic antinomies were discovered: it turned out that when using set-theoretic representations, some statements can be proved together with their negations (and then, according to the rules of classical propositional logic, absolutely any statement can be “proved!”). The antinomies marked the complete failure of Cantor's program.

The culprit of the failure was none other than Bertrand Russell.
However, this theory managed to completely take over the minds of contemporaries.

Here is what David Hilbert writes about Cantor and his Mengenlehre (which I have already talked about here):

No one will ever drive us out of his paradise.
(c) David Hilbert. In defense of Cantor's theory of sets.

Georg Cantor (photo is given later in the article) is a German mathematician who created set theory and introduced the concept of transfinite numbers, infinitely large, but different from each other. He also defined ordinal and cardinal numbers and created their arithmetic.

Georg Kantor: a short biography

Born in St. Petersburg on 03/03/1845. His father was a Dane of the Protestant faith, Georg-Valdemar Kantor, who was engaged in trade, including on the stock exchange. His mother Maria Bem was a Catholic and came from a family of prominent musicians. When Georg's father fell ill in 1856, the family moved first to Wiesbaden and then to Frankfurt in search of a milder climate. The boy's mathematical talents showed up even before his 15th birthday while studying at private schools and gymnasiums in Darmstadt and Wiesbaden. In the end, Georg Cantor convinced his father of his firm intention to become a mathematician, not an engineer.

After a short study at the University of Zurich, in 1863 Kantor transferred to the University of Berlin to study physics, philosophy and mathematics. There he was taught:

• Karl Theodor Weierstrass, whose specialization in analysis probably greatest influence on George;
• Ernst Eduard Kummer, who taught higher arithmetic;
• Leopold Kronecker, number theorist who later opposed Cantor.

After spending one semester at the University of Göttingen in 1866, the following year Georg wrote a doctoral dissertation entitled "In mathematics the art of asking questions is more valuable than solving problems", concerning a problem that Carl Friedrich Gauss left unsolved in his Disquisitiones Arithmeticae (1801) . After briefly teaching at the Berlin School for Girls, Kantor began working at the University of Halle, where he remained until the end of his life, first as a teacher, from 1872 as an assistant professor, and from 1879 as a professor.

Research

At the beginning of a series of 10 papers from 1869 to 1873, Georg Cantor considered number theory. The work reflected his passion for the subject, his studies of Gauss and the influence of Kronecker. At the suggestion of Heinrich Eduard Heine, Cantor's colleague in Halle, who recognized his mathematical talent, he turned to the theory of trigonometric series, in which he expanded the concept of real numbers.

Based on the work on the function of a complex variable by the German mathematician Bernhard Riemann in 1854, in 1870 Kantor showed that such a function can be represented in only one way - by trigonometric series. The consideration of a set of numbers (points) that would not contradict such a representation led him, firstly, in 1872 to a definition in terms of rational numbers (fractions of integers) and then to the beginning of work on his life's work, set theory and the concept transfinite numbers.

set theory

Georg Cantor, whose set theory originated in correspondence with the mathematician of the Technical Institute of Braunschweig Richard Dedekind, was friends with him since childhood. They came to the conclusion that sets, whether finite or infinite, are collections of elements (for example, numbers, (0, ±1, ±2 . . .)) that have a certain property while retaining their individuality. But when Georg Cantor used a one-to-one correspondence (for example, (A, B, C) to (1, 2, 3)) to study their characteristics, he quickly realized that they differ in the degree of their membership, even if they were infinite sets , i.e. sets, a part or subset of which includes as many objects as it itself. His method soon gave amazing results.

In 1873 Georg Cantor (mathematician) showed that rational numbers, although infinite, are countable because they can be put in one-to-one correspondence with natural numbers (i.e. 1, 2, 3, etc.). He showed that the set of real numbers, consisting of irrational and rational ones, is infinite and uncountable. More paradoxically, Cantor proved that the set of all algebraic numbers contains as many elements as the set of all integers, and that the non-algebraic transcendental numbers, which are a subset of irrational numbers, are uncountable and therefore more numerous than integers. , and should be treated as infinite.

Opponents and supporters

But Kantor's paper, in which he first put forward these results, was not published in the journal Krell, since one of the reviewers, Kronecker, was categorically against it. But after the intervention of Dedekind, it was published in 1874 under the title "On characteristic properties all real algebraic numbers.

Science and personal life

In the same year, during his honeymoon with his wife Valli Gutman, Kantor met Dedekind, who spoke favorably of his new theory. George's salary was small, but with the money of his father, who died in 1863, he built a house for his wife and five children. Many of his papers were published in Sweden in the new journal Acta Mathematica, edited and founded by Gesta Mittag-Leffler, who was among the first to recognize the talent of the German mathematician.

Connection with metaphysics

Cantor's theory became an entirely new subject of study concerning the mathematics of the infinite (eg series 1, 2, 3, etc., and more complex sets), which depended heavily on one-to-one correspondence. The development by Cantor of new methods for posing questions concerning continuity and infinity gave his research an ambiguous character.

When he argued that infinite numbers really exist, he turned to ancient and medieval philosophy regarding actual and potential infinity, as well as to the early religious education that his parents gave him. In 1883, in his book Foundations of General Set Theory, Cantor combined his concept with Plato's metaphysics.

Kronecker, who claimed that only integers “exist” (“God created the integers, the rest is the work of man”), for many years ardently rejected his reasoning and prevented his appointment at the University of Berlin.

transfinite numbers

In 1895-97. Georg Cantor fully formed his notion of continuity and infinity, including infinite ordinal and cardinal numbers, in his most famous work, published as Contributions to the Establishment of the Theory of Transfinite Numbers (1915). This essay contains his concept, to which he was led by demonstrating that an infinite set can be put in a one-to-one correspondence with one of its subsets.

By the least transfinite cardinal number, he meant the cardinality of any set that can be put in a one-to-one correspondence with the natural numbers. Cantor called it aleph-null. Large transfinite sets are denoted aleph-one, aleph-two, etc. He further developed the arithmetic of transfinite numbers, which was analogous to finite arithmetic. Thus, he enriched the concept of infinity.

The opposition he encountered, and the time it took for his ideas to be fully accepted, are explained by the difficulty of re-evaluating the ancient question of what a number is. Cantor showed that the set of points on a line has a higher cardinality than aleph-zero. This led to the well-known problem of the continuum hypothesis - there are no cardinal numbers between aleph-zero and the power of points on the line. This problem in the first and second half of the 20th century aroused great interest and was studied by many mathematicians, including Kurt Gödel and Paul Cohen.

Depression

The biography of Georg Kantor since 1884 was overshadowed by his mental illness, but he continued to work actively. In 1897 he helped hold the first international mathematical congress in Zurich. Partly because he was opposed by Kronecker, he often sympathized with young novice mathematicians and sought to find a way to save them from the harassment of teachers who felt threatened by new ideas.

Confession

At the turn of the century, his work was fully recognized as the basis for function theory, analysis, and topology. In addition, the books of Cantor Georg served as an impetus for further development intuitionist and formalistic schools of the logical foundations of mathematics. This significantly changed the teaching system and is often associated with the "new mathematics".

In 1911, Kantor was among those invited to the celebration of the 500th anniversary of the University of St. Andrews in Scotland. He went there in the hope of meeting with whom, in his recently published Principia Mathematica, he repeatedly referred to the German mathematician, but this did not happen. The university awarded Kantor an honorary degree, but due to illness, he was unable to accept the award in person.

Kantor retired in 1913, lived in poverty and went hungry during the First World War. Celebrations in honor of his 70th birthday in 1915 were canceled due to the war, but a small ceremony took place at his home. He died on 01/06/1918 in Halle, in a psychiatric hospital, where he spent last years own life.

Georg Kantor: biography. Family

On August 9, 1874, the German mathematician married Wally Gutman. The couple had 4 sons and 2 daughters. The last child was born in 1886 in a new house purchased by Kantor. His father's inheritance helped him support his family. Kantor's state of health was strongly affected by the death of his youngest son in 1899 - since then depression has not left him.

set theory A branch of mathematics that studies the general properties of sets. Set theory is at the heart of most mathematical disciplines; it had a profound effect on the understanding of the subject matter of mathematics itself.
Georg Cantor (1845 - † 1918), founder of set theory Until the second half of the 19th century, the concept of "set" was not considered as mathematical ("a lot of books on a shelf", "a lot of human virtues", etc. - all these are purely everyday turns) . The situation changed when the German mathematician Georg Cantor developed his program for the standardization of mathematics, in which any mathematical object had to be one or another “set”. For example, a natural number, according to Cantor, should be considered as a set consisting of a single element of another set, called the "natural series", which, in turn, is a set that satisfies the so-called Peano axioms. Wherein general concept"set", which he considered as central to mathematics, Kantor gave very vague definitions, such as "a set is a lot, conceivable as a single", etc. This was quite consistent with the mindset of Cantor himself, which emphatically called his program not "the theory of sets ” (this term appeared much later), but “the doctrine of sets” (Mengenlehre).
Cantor's program provoked strong protests from many of his contemporaries. famous mathematicians. Leopold Kronecker was especially distinguished by his irreconcilable attitude towards it, who believed that only integers and what directly boils down to them (his phrase is known that “God created natural numbers, and everything else is the work of human hands”). However, some other mathematicians - in particular, Gottlob Frege and David Hilbert - supported Cantor in his intention to translate all of mathematics into a set-theoretic language.
However, it soon became clear that Cantor's direction to unlimited arbitrariness when operating with sets (expressed by himself in the principle "the essence of mathematics lies in its freedom") was initially imperfect, namely, a number of set-theoretic antinomies were discovered: it turned out that when using the theoretical multiple representations, some statements can be proved together with their negations (and then, according to the rules of classical propositional logic, absolutely any statement can be “proved”). The antinomies marked the complete failure of Cantor's program.
In the early 20th century, Bertrand Russell, while studying naive set theory, came up with a paradox (since known as Russell's paradox). In this way, the inconsistency of naive set theory and Cantor's program for the standardization of mathematics associated with it was demonstrated.
After the discovery of Russell's antinomy, some mathematicians (for example, L. E. Ya. Brouwer and his school) decided to completely abandon the use of set-theoretic representations. The other part of mathematicians, headed by D. Hilbert, made a number of attempts to substantiate that part of the set-theoretic representations, which seemed to them the most responsible for the emergence of antinomies, on the basis of obviously reliable finite mathematics. To this end, various axiomatizations of set theory have been developed.
A feature of the axiomatic approach is the rejection of the idea of ​​the actual existence of sets in some ideal world embedded in Cantor's program. Within the framework of axiomatic theories, sets "exist" in a purely formal way, and their "properties" may depend significantly on the choice of axiomatics. This fact has always been the target of criticism from those mathematicians who did not agree (as Hilbert insisted) to recognize mathematics as a game of symbols devoid of any content. In particular, N. N. Luzin wrote that “the power of the continuum, if only to think of it as a set of points, is a certain single reality”, the place of which in the series of cardinal numbers cannot depend on whether the continuum hypothesis is recognized as an axiom, or its negation .
Now the most common axiomatic set theory is ZFC - the Zermelo-Fraenkel theory with the axiom of choice. The question of the consistency of this theory (and even more so, the existence of a model for it) remains unresolved.
Set theory is based on primary concepts: a bunch of and attitude be an element sets (denoted as - "x is an element of the set A"). Among the derived concepts, the most important are the following:
The following operations are defined on sets:
The following binary relations are defined for sets: