Cantor's paradox, also known as the "maximum cardinal number paradox", isset theoryOne of the paradoxes.By the founder of set theory and German mathematiciancantor It was proposed in 1899.Consider everythingaggregateLet the cardinal number of the set V formed be λ.Since V is the largest set, λ should be the largestbase, but set theoreticCantor theoremIt is known that the power set of each set has a greater cardinality than that of the set, so the power set of V will have a greater cardinality than V, which contradicts with λ being the maximum cardinality.[1]
The annual art festival of a middle school is coming again.The festival has three events: calligraphy and painting competition, singing competition and go competition.Meng Juan, a member of the Literature and Art Committee of Class 2 and Class 3 of Junior High School, counted the participants of this class. The result was: 15 people participated in the calligraphy and painting competition, 28 people participated in the singing competition, and 25 people participated in the Go competition. Meng Juan was puzzled that the total number of participants was 68, while there were only 60 people in her class. Where did the remaining 8 people come from?It turns out that this is caused by the nature of the set.
Set theory
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The theory of set came into being at the end of the 19th century.At that time, German mathematician Cantor tried to answer some mathematical problems involving infinite quantities, such as "How many integers are there?" "How many points are there on a circle?" Are there more points between 0 and 1 than on a 1-inch line segment? "wait.And“integer”, "points on the circumference", "numbers between 0 and 1", etc. are all sets, so the study of these problems produces set theory.
What is a collection?In Cantor's words, a set is a whole that gathers some specific or ideological objects that are different from each other.In short, a set is a group of things.For example, "municipalities directly under the Central Government of the People's Republic of China", "people who are late for math class on Tuesday", "Zhang San's shoes", etc. are all collections.Birds of a feather flock together. People or things of the same kind always have the same characteristics or properties. According to these characteristics or properties, a class can be determined. This class is a collection.
The things that constitute a set belong to this set, and the individuals that belong to this set are called the elements of the set. For example, "positive odd numbers less than 7" are a set, and the 1, 3, and 5 that constitute this set are the elements of this set.Given a set, it defines which elements the set is composed of.Obviously, for anything, it either belongs to a set or does not belong to this set, and the two must be in one place.For example, 1 and 3 belong to the set of "positive odd numbers less than 7", while 6 and 8 do not belong to this set.
In arithmetic, we often compare some numbers to find out which one is larger.Collections can also be compared, and one method of comparison is to compare the elements of one collection with those of another.The set {1,3,5,7} is different from the set {2,4,6,8} because their elements are different.The set A={a, b, c} is the same as the set B={c, b, a}, because these two sets have the same elements. In this case, we call them A=B.It doesn't matter whether the elements are arranged in the same order. As long as two sets have the same elements, they are equal.
The one-to-one correspondence method can also be used to compare sets.In ancient times, a man was framed and put into a dark basement. He wanted to go out for revenge as soon as possible. But in this dark world, there is no difference between night and day, and certainly no concept of days. How can he know how many days he has stayed here?He found a trick. The jailer emptied the toilet every other day.So whenever the jailer empties the toilet, he draws a line on the wall with a stone, so that the collection of toilets and the collection of lines form a one-to-one correspondence, and the collection of toilets and the collection of dates form a one-to-one correspondence, so the number of days can be known from the number of lines.
To compare any two sets, just use the elements of one set to correspond to the elements of the other set.If there is a one-to-one correspondence between two sets, then we say that the two sets are equivalent. For example, the set of lines, the set of toilets, and the set of dates are equivalent to each other.But it is worth noting that the equivalence and equality of two sets are not the same thing.For example, there are two students, Zhang San and Li Si, in Class 1 and Class 2 of Junior High School. The set A of Zhang San's teacher is equal to the set B of Li Si's teacher, because the elements of the two sets are identical;That is:
A={Wang Wu, Zhao Liu, Zhou Qi}
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B={Wang Wu, Zhao Liu, Zhou Qi}
But if Zhang San and Li Si are not from the same school, the set A of Zhang San's teachers and the set C of Li Si's teachers are not equal but can be equivalent, because the elements of the two sets are only one-to-one correspondence, not the same, that is:
A={Wang Wu, Zhao Liu, Zhou Qi}
C={Wu Ba, Zheng Jiu, Chen Shi}
The simplest way to judge whether several sets are equivalent is to see whether the number of elements in each set is equal. The number of elements in a set is called the cardinal number of the set. For example, {Beijing, Tianjin, Shanghai} has three elements, so its cardinal number is 3. {Kong Yiji, Wind Wave, True Story of Ah Q, A Little Thing} has four elements, so its cardinal number is 4.
There are some sets whose elements are finite, such as {1, 4, 9,... 100}, {Reagan, Bush, Clinton}. Such sets are called finite sets.Some sets have infinite elements, such as the set of integers, the set of stars in the universe, etc. This set is called infinite set.The cardinality of an infinite set is greater than that of any finite set.From the analysis in the previous section, we can see that infinite sets can be compared by one-to-one correspondence, but there are surprising results. For example, even sets have as many elements as natural sets, and the set of points on a line is equal to the set of points on a plane.Cantor took the concept of infinite set as the basis of set theory, and proved that a remarkable feature of infinite set is that infinite set itself can have one-to-one correspondence with its parts.
Another kind of set is just the opposite of infinite set, which does not contain any elements, such as "odd sets that can be divided by 2", "sets of people who live to 1200 years old", etc. These sets are called empty sets.When we discuss objects with certain properties, we call the set composed of all elements with such common properties the complete set.For example, in a certain sports meeting, there are 10 athletes participating in a certain event, so the collection of these 10 athletes is the complete collection of athletes.
In a set, we can take out some elements to form a new set.In the example mentioned at the beginning of this section, "students in Class 2 · 3 of Junior High School" is a collection, and among these students, several different types of students can be distinguished, such as students who participate in singing competitions, students who participate in calligraphy and painting competitions, and students who participate in Go competitions.These kinds of students are several collections made up of students in Class 3, Grade 2 of Junior High School. These collections are subsets of students in Class 3, Grade 2 of Junior High School.Obviously, a subset is an element contained in the subset of the original set. For example, Zhang San is not only an element of the student set participating in the calligraphy and painting competition, but also an element of the student set of Class 3, Grade 2.Of course, we can also form different subsets according to other conditions.Such as male student group, female student group, League member student group, student group participating in English learning, etc.How many subsets can a given set form?Let's take a closer look, for example:
{1,2,3} can have {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} 8 subsets.
By analogy, we can see that a set with n elements has 2 ^ n subsets.
It should be noted that in set theory, there is no limit on the number of elements in a set, so there will be a subset (empty set) with only one element or no element. The original set itself is also its own subset.So when we ask how many subsets the original set can have, the empty set and the original set must also be included.
All subsets of a set can also form a set, which is called the power set of the original set.For example, the power set of {Zhang 3, Li 4} is
{{} {Zhang 3}, {Li 4}, {Zhang 3, Li 4}}.
Two or more sets can also form new sets through operations.For example, A={Zhao Li, Wang Fang, Chen Feng} for students with outstanding English tests, and B={Zhu Jun, Wang Ming, Wang Fang} for students with outstanding math tests.These two sets can be added to form set C, which contains both elements of A and B, and this set is {Zhao Li, Wang Fang, Chen Feng, Zhu Jun, Wang Ming}.This set is called the union of A and B.It should be noted that Wang Fang does not need to write twice in the above collection. Just writing once means that she is an element of C.Therefore, the cardinality of C is not equal to the cardinality of A plus the cardinality of B, but the common elements are subtracted after the two are added.Yu Juan, a member of the literature and art committee, actually adds three subsets when making statistics, but only by adding the cardinals of the three subsets and subtracting the common elements can the total number of people in Class 2 and Class 3 be equal to.Meng Juan simply adds up and forgets to subtract the same elements. No wonder there are 8 more people.The set A and B can also be multiplied to get a new set D, because the set consisting of the common elements in A and B is {Wang Fang}, and D is called the intersection of A and B.
These are some basic concepts of Cantor's set theory.At that time, the attack of Kronecker, the German mathematical authority and his teacher, was particularly fierce.He said, "Cantor has gone into the hell of super poor numbers." He had a famous saying: "God created positive integers, and the rest is the work of people." That is to say, people can onlypositive integerAs for the infinite world, it is totally beyond human ability.He didn't even recognize Cantor as his student.In this case, Cantor was suppressed and excluded for a long time, and he could not get the professorship of Berlin University. He was depressed and frustrated. He had a mental breakdown, gave up mathematical research, and finally died in a psychiatric hospital.
However, the establishment of Cantor's set theory is a milestone in the history of human thinking, which marks that after thousands of years of efforts, human beings have finally basically understood the nature of infinity.Therefore, more and more people begin to recognize it and successfully apply it to many other mathematical fields.It is believed that set theory is indeed the foundation of mathematics.Moreover, due to the establishment of set theory, the "absolute strictness" of mathematics has been achieved.At this time, the kingdom of mathematics was full of bright spring, warm sunshine and peaceful scenes.However, just when people were jubilantly preparing for the "Hundred Bulls Feast", an unprecedented strong earthquake suddenly broke out on the land of the mathematical kingdom - a series of paradoxes were found in set theory.
The appearance of these paradoxes can be said to be the inevitable result of Cantor's set theory.In fact, at the end of the 19th century, Cantor himself had found many contradictions in his theory, but he did not make a statement, but quietly used them.
It can be seen from the above that there are 2 subsets of a set with 1 element, and there are 4 subsets of a set with 2 elements. Generally, there are 2 ^ n subsets of a set with n elements, and the cardinality of a set with n elements is n, while the cardinality of a set composed of all its subsets is 2 ^ n, obviously 2 ^ n>n.So there are“Cantor theorem”: Of the power set of any set (including infinite set)baseGreater than the cardinality of this arbitrary set.
Theoretical influence
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According to Cantor's set theory, any property can determine a set, so that all sets can form a set, that is, the "set of all sets" (the great complete set).Obviously, this set should be the largest set, so its cardinality should also be the largest. However, the cardinality of its subset set is“Cantor theorem”It must be bigger, then, "the set of all sets" cannot be "the set of all sets", which is "Cantor paradox".Cantor was not afraid of this paradox, because he just proved that there is no "set of all sets" or "largest set", and of course there is no "largest cardinality" through the method of contradiction.
The appearance of the paradox did not cause much shock at this time. People felt that it only seemed to involve some technical problems of the set theory. As long as appropriate corrections were made, the set theory would still become the foundation of the mathematical mansion. Cantor only used the paradox to make counterevidence, but did not study the source and significance of the paradox carefully. He did not realize why such counterevidence was possible,Because the basic concepts "set", "belong" and "element" used in his theory contain contradictions.Russell's“Russell paradox”Then "the technical details of mathematics have been stripped off", exposing the contradictions in it naked!
paradox
《Ancient and modern mathematical thought》In the book (Volume IV, page 289), it is pointed out that the most in-depth activities in mathematics in the twentieth century make the discussion of the foundation, the problems imposed on mathematicians, and the problems they voluntarily assume involve not only the essence of mathematics, but also the correctness of deductive mathematics.
In the early part of this century, several activities converged to lead the basic problems to a climax. The first is the discovery of contradictions, euphemistically known asparadox, especially in set theory.
