Is mathematics as accurate as it seems?
Probably, this question was asked by everyone, a little bit interested in mathematics. After reading the article 2 x 2 = 4 , it was concluded that this topic may also appeal to habra people. It will be about axioms in mathematics, contradictions and paradoxes. Who cares - welcome under cat.
Each of us at school did not doubt the validity of certain mathematical statements. Well, it is true that the teacher said, then the truth. But, having got acquainted with strict mathematics (I do not like the word “higher”), we began to understand that the more we try to formalize the subject, the more difficult it is to do, and sometimes it does not work at all.
So to us, the usual real numbers, for Leopold Kronecker, were not such, he said: “God created natural numbers, and everything else is the work of human hands” (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”)
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After Georg Cantor proved that the segment is equally powerful (A and B are equally powerful if there is a bijection between them) to n-dimensional space, he proclaimed: “I see it, but I do not believe it!” (“Je le vois, mais je ne le crois pas! ”)
This article focuses on the axiomatics of various mathematical sets, operations, etc., but it will still be a logical question, why do we need axioms? I will give a simple example. Take the Russian language and the word, for example, "deja vu". Let's look at its meaning, “Deja vu is a mental state in which a person feels that he has once been in a similar situation.” But we are meticulous, therefore, now instead of one word in front of us there will be much more. What is "mental", "state", "person", "feel", "similar", "situation". As you can see, we have a tree of words, and due to the fact that there are a lot of words that have a meaning in Russian, we will have a path in the tree where the same word occurs twice, i.e. we have identified it through ourselves.
That's what axioms are for. We always need a foundation from which we can start, something that is intuitively clear to everyone. Inaccuracy 1. In mathematics, there are often statements that are intuitive, but lead to paradoxes. For example, the axiom of choice ( Axiom of Choice ), but we'll talk about this a little later.
I, as a programmer, love to assume that 0 belongs to natural numbers, it is convenient. Well, now Peano’s most famous axiom.
1. 0 is a natural number.
2. The number following the natural is also natural.
3. 0 does not follow any natural number.
4. If a positive integer a immediately follows both b and c , then b and c are the same.
5. (Induction axiom) If a sentence is proved for 0 (the base of induction) and if the assumption that it is true for a natural number n , it follows that it is true for the natural number following n (induction hypothesis), then this sentence true for all natural numbers.
We will understand in order.
The 1st axiom says that there is at least one natural number. Otherwise, we would say that this is generally an empty set and all axioms would be fulfilled for it.
2 and 3 seem to be so clear.
4. This axiom is needed in order to avoid “branches”. Otherwise, we could say that 3 follows 2 and 2 ', and then 2 and 2' after 1 and 1 ', respectively, etc. In principle, such a model has the right to exist, but it is extremely difficult to introduce an order relation on it.
5. The first person in line is a woman. For every woman goes a woman. In real life, this means that the whole line consists of women. And since we want to describe more vital objects, we introduce the axiom of induction, for it does not follow from the preceding ones.
Comfortable model, everything is fine, everyone is happy. The question is, what's the catch? It turns out that if we add a new natural number with to our usual natural numbers and say that it is larger than all our usual numbers, then we will not come to any contradiction. Those. we have not only our model N, but also, for example, N + Z. Where in N and Z (integers) is the usual comparison of numbers, as well as any number from N is less than any number from Z.
The question is, is it possible to introduce axioms so that we describe our usual natural numbers, and only them (ie, is there a formula, substituting into which the natural number it gives True, and any other number False)? The answer is no. The idea of ​​the proof is that all formulas can be encoded with natural numbers. And then, by writing a cunning formula, and substituting its code in F (a formula that by assumption is able to determine natural naturalness), we get a contradiction.
Modern mathematics is being built on the axioms below, well, take a deep breath ... let's get started. To begin with, I’ll make a reservation that we will consider all sorts of sets. For example, the set of all houses in Russia, at the same time, each element of the set, in this case a house, may contain some more sets, it may well be that they turned out to be heterogeneous, for example, the number of routers in a house (the unit is the number ) and people (units - people) living in this house. A more natural example for programmers is nested lists [[1, 2, [3, -19]], [0, 1], [5, [26, 1]], 27]. In this example, we have a set consisting of 4 elements [1, 2, [3, -19]], [0, 1], [5, [26, 1]], 27. For a clear awareness, we note that 0 is not an element of this set, although if you dig in depth, it turns out that 0 is there! We now turn to the axioms. I allow myself not to give tedious language, but to explain in my own words.
1. The axiom of bulk. If two sets consist of identical elements, then they are equal.
2. Axiom of subsets. If we have some formula, then from any set it “cuts out” also set.
3. The axiom of replacement. If for each plurality of x , F (x) = {y | F (x, y)} is also a set, then for any a , {y | x belongs to a, y belongs to f (x)} - also a set.
4. Axiom of degree. The set of subsets is also a set.
5. The axiom of infinity. There is a set that contains an empty set, and also along with each element x contains the set {{x}, x} - i.e. all elements x and x itself as an element.
6. The axiom of regularity. There are no infinite chains of sets, i.e. it is impossible for a set a1 to contain a2 , then in turn a3 , etc.
Explanations.
1. Everything is clear.
2. Let we consider the set of natural numbers. And the formula is: x! = 0. It is clear that all natural numbers except zero satisfy it. The axiom says that natural numbers without zero are also many. If you try to generalize this theorem, you get the Russell paradox .
3. I did not know how simpler it would be to describe this axiom, in a nutshell, if we combine the sets, we get a set.
4. [1, 2, 3], the set of subsets of this set (I apologize for the large specific density of the word “set”) - 1, 2, 3, [1, 2], [1, 3], [2, 3] , [1, 2, 3]. The question is, when can it happen that we will do something and we will not have many? Well, at least consider the set of all sets! According to axiom 4, there are many of its subsets, and it is not difficult to prove that it is more powerful than our pluralism.
5. The first axiom, where the existence of at least some set is asked. Which one is described in the axiom.
6. Everything is clear too.
Well, done with nudyatiny. With the help of these axioms one can construct natural numbers, for example. They will look like this (e is an empty set). 0 = e, 1 = {e}, 2 = {e, {e}}, 3 = {e, {e}, {e, {e}}}, etc. As a matter of fact, modern mathematics is built on this theory.
Firstly, it is not proved that the axioms of ZF are consistent, if they are contradictory, then we can derive any statement, for example, 0 = 1, and our science is worthless. Even more, it has been proven that it is impossible to prove the consistency of ZF. A funny thing happens, but there's nothing to worry about. If we cannot prove something, it does not mean that it is not, in this case, consistency. Moving on.
Mathematics turns out to be a rather stingy science, that is, little can be proved if you do not add the axiom of choice. And what is this axiom? In three words - from any non-empty set, you can select an element. It would seem to be a very natural axiom, but it leads to the Banach-Tarski paradox , which means that the ball can be broken into 5 pieces and 2 of them can be assembled from them. Those. Apple can be cut into 5 pieces and collect two apples? Therefore, the paradox. What is even more interesting, it is proved that if the theory of ZF is consistent, then adding to it the axiom of choice (ZF + Axiom of Choice = ZFC) we get a consistent axiom!
That we can not prove something, then some paradoxes. Maybe mathematics is complete nonsense? Maybe you should not study it? Answer: no nonsense, should study. Why, the reader will ask. I will give enough physical evidence. Usually it happens in physics. "Wow, for 100 years we have watched the fall of sandwiches and it turned out that they are falling buttered down, let's call it the law." Think kidding? And try to prove that the bodies are made up of molecules. Nothing more rigorous than the fact that for 2000 years this theory has not failed, you will not think up. So with the mathematics about the same situation. We use it, it seems that cars are going, planes are flying, buildings are standing and everything is fine. It is intuitively clear that if there was a contradiction in mathematics, the deeper we would dig in the wilds of this science, the easier it would be to prove theorems, but this does not happen.
And yet, from where the Banach-Tarsk paradox arises, it’s still quite logical! In fact, if you carefully note, then in the universe there is nothing infinite. There is nothing infinitely small, etc. It is just convenient to work with infinite sets. So it is quite normal that results not applicable to life can be obtained.
Good luck to everyone in the study of this subject! =)
Instead of the preface
Each of us at school did not doubt the validity of certain mathematical statements. Well, it is true that the teacher said, then the truth. But, having got acquainted with strict mathematics (I do not like the word “higher”), we began to understand that the more we try to formalize the subject, the more difficult it is to do, and sometimes it does not work at all.
So to us, the usual real numbers, for Leopold Kronecker, were not such, he said: “God created natural numbers, and everything else is the work of human hands” (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”)
')
After Georg Cantor proved that the segment is equally powerful (A and B are equally powerful if there is a bijection between them) to n-dimensional space, he proclaimed: “I see it, but I do not believe it!” (“Je le vois, mais je ne le crois pas! ”)
A little philosophy
This article focuses on the axiomatics of various mathematical sets, operations, etc., but it will still be a logical question, why do we need axioms? I will give a simple example. Take the Russian language and the word, for example, "deja vu". Let's look at its meaning, “Deja vu is a mental state in which a person feels that he has once been in a similar situation.” But we are meticulous, therefore, now instead of one word in front of us there will be much more. What is "mental", "state", "person", "feel", "similar", "situation". As you can see, we have a tree of words, and due to the fact that there are a lot of words that have a meaning in Russian, we will have a path in the tree where the same word occurs twice, i.e. we have identified it through ourselves.
That's what axioms are for. We always need a foundation from which we can start, something that is intuitively clear to everyone. Inaccuracy 1. In mathematics, there are often statements that are intuitive, but lead to paradoxes. For example, the axiom of choice ( Axiom of Choice ), but we'll talk about this a little later.
More specifics. Peano's axioms of natural numbers.
I, as a programmer, love to assume that 0 belongs to natural numbers, it is convenient. Well, now Peano’s most famous axiom.
1. 0 is a natural number.
2. The number following the natural is also natural.
3. 0 does not follow any natural number.
4. If a positive integer a immediately follows both b and c , then b and c are the same.
5. (Induction axiom) If a sentence is proved for 0 (the base of induction) and if the assumption that it is true for a natural number n , it follows that it is true for the natural number following n (induction hypothesis), then this sentence true for all natural numbers.
We will understand in order.
The 1st axiom says that there is at least one natural number. Otherwise, we would say that this is generally an empty set and all axioms would be fulfilled for it.
2 and 3 seem to be so clear.
4. This axiom is needed in order to avoid “branches”. Otherwise, we could say that 3 follows 2 and 2 ', and then 2 and 2' after 1 and 1 ', respectively, etc. In principle, such a model has the right to exist, but it is extremely difficult to introduce an order relation on it.
5. The first person in line is a woman. For every woman goes a woman. In real life, this means that the whole line consists of women. And since we want to describe more vital objects, we introduce the axiom of induction, for it does not follow from the preceding ones.
Comfortable model, everything is fine, everyone is happy. The question is, what's the catch? It turns out that if we add a new natural number with to our usual natural numbers and say that it is larger than all our usual numbers, then we will not come to any contradiction. Those. we have not only our model N, but also, for example, N + Z. Where in N and Z (integers) is the usual comparison of numbers, as well as any number from N is less than any number from Z.
The question is, is it possible to introduce axioms so that we describe our usual natural numbers, and only them (ie, is there a formula, substituting into which the natural number it gives True, and any other number False)? The answer is no. The idea of ​​the proof is that all formulas can be encoded with natural numbers. And then, by writing a cunning formula, and substituting its code in F (a formula that by assumption is able to determine natural naturalness), we get a contradiction.
More specifics. Axiomatics of Zermelo-Fraenkel sets (ZF)
Modern mathematics is being built on the axioms below, well, take a deep breath ... let's get started. To begin with, I’ll make a reservation that we will consider all sorts of sets. For example, the set of all houses in Russia, at the same time, each element of the set, in this case a house, may contain some more sets, it may well be that they turned out to be heterogeneous, for example, the number of routers in a house (the unit is the number ) and people (units - people) living in this house. A more natural example for programmers is nested lists [[1, 2, [3, -19]], [0, 1], [5, [26, 1]], 27]. In this example, we have a set consisting of 4 elements [1, 2, [3, -19]], [0, 1], [5, [26, 1]], 27. For a clear awareness, we note that 0 is not an element of this set, although if you dig in depth, it turns out that 0 is there! We now turn to the axioms. I allow myself not to give tedious language, but to explain in my own words.
1. The axiom of bulk. If two sets consist of identical elements, then they are equal.
2. Axiom of subsets. If we have some formula, then from any set it “cuts out” also set.
3. The axiom of replacement. If for each plurality of x , F (x) = {y | F (x, y)} is also a set, then for any a , {y | x belongs to a, y belongs to f (x)} - also a set.
4. Axiom of degree. The set of subsets is also a set.
5. The axiom of infinity. There is a set that contains an empty set, and also along with each element x contains the set {{x}, x} - i.e. all elements x and x itself as an element.
6. The axiom of regularity. There are no infinite chains of sets, i.e. it is impossible for a set a1 to contain a2 , then in turn a3 , etc.
Explanations.
1. Everything is clear.
2. Let we consider the set of natural numbers. And the formula is: x! = 0. It is clear that all natural numbers except zero satisfy it. The axiom says that natural numbers without zero are also many. If you try to generalize this theorem, you get the Russell paradox .
3. I did not know how simpler it would be to describe this axiom, in a nutshell, if we combine the sets, we get a set.
4. [1, 2, 3], the set of subsets of this set (I apologize for the large specific density of the word “set”) - 1, 2, 3, [1, 2], [1, 3], [2, 3] , [1, 2, 3]. The question is, when can it happen that we will do something and we will not have many? Well, at least consider the set of all sets! According to axiom 4, there are many of its subsets, and it is not difficult to prove that it is more powerful than our pluralism.
5. The first axiom, where the existence of at least some set is asked. Which one is described in the axiom.
6. Everything is clear too.
Well, done with nudyatiny. With the help of these axioms one can construct natural numbers, for example. They will look like this (e is an empty set). 0 = e, 1 = {e}, 2 = {e, {e}}, 3 = {e, {e}, {e, {e}}}, etc. As a matter of fact, modern mathematics is built on this theory.
Contradictions and paradoxes
Firstly, it is not proved that the axioms of ZF are consistent, if they are contradictory, then we can derive any statement, for example, 0 = 1, and our science is worthless. Even more, it has been proven that it is impossible to prove the consistency of ZF. A funny thing happens, but there's nothing to worry about. If we cannot prove something, it does not mean that it is not, in this case, consistency. Moving on.
Mathematics turns out to be a rather stingy science, that is, little can be proved if you do not add the axiom of choice. And what is this axiom? In three words - from any non-empty set, you can select an element. It would seem to be a very natural axiom, but it leads to the Banach-Tarski paradox , which means that the ball can be broken into 5 pieces and 2 of them can be assembled from them. Those. Apple can be cut into 5 pieces and collect two apples? Therefore, the paradox. What is even more interesting, it is proved that if the theory of ZF is consistent, then adding to it the axiom of choice (ZF + Axiom of Choice = ZFC) we get a consistent axiom!
A spark of hope
That we can not prove something, then some paradoxes. Maybe mathematics is complete nonsense? Maybe you should not study it? Answer: no nonsense, should study. Why, the reader will ask. I will give enough physical evidence. Usually it happens in physics. "Wow, for 100 years we have watched the fall of sandwiches and it turned out that they are falling buttered down, let's call it the law." Think kidding? And try to prove that the bodies are made up of molecules. Nothing more rigorous than the fact that for 2000 years this theory has not failed, you will not think up. So with the mathematics about the same situation. We use it, it seems that cars are going, planes are flying, buildings are standing and everything is fine. It is intuitively clear that if there was a contradiction in mathematics, the deeper we would dig in the wilds of this science, the easier it would be to prove theorems, but this does not happen.
And yet, from where the Banach-Tarsk paradox arises, it’s still quite logical! In fact, if you carefully note, then in the universe there is nothing infinite. There is nothing infinitely small, etc. It is just convenient to work with infinite sets. So it is quite normal that results not applicable to life can be obtained.
Good luck to everyone in the study of this subject! =)
Source: https://habr.com/ru/post/167583/
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