Finite and infinite in mathematics. Lecture by Pavel Kozhevnikov for high school students in Yandex
Unlike the world around us, in which everything is finite, in mathematics we often encounter infinite objects. For example, infinite sets of integers, rational, algebraic, constructive, or real numbers. At the lecture, we will consider problems in which some principles of work with infinite sets appear. Sometimes these principles are very different from those to which we are accustomed in the case of the "finite" world.
Imagine a hypothetical hotel with an infinite number of rooms. All hotel rooms are already settled. but at some point another guest arrives at the hotel. Despite the fact that every hotel room already has at least one tenant, you can still find a room for a new one. After all, our hotel is infinite, and if you ask all the guests to move one room further, the very first room will be free. And even if several new guests arrive, by a series of similar relocations, it will be possible to find a place for them.
Now imagine that we have three such Hilbert hotels. Two of them are closed for repairs, and guests need to settle in one remaining. Even such an unusual case should not cause any problems, all guests will find a place in the third infinite hotel. For example, we can consistently resettle guests from two hotels on even and odd numbers, respectively.
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Imagine an even more strange situation: an infinite number of Hilbert hotels are being closed for renovation, and we need to settle all their guests in one endless hotel. If we use the technique from the previous example, but we will settle the guests consistently diagonally, sooner or later we will be able to distribute all.
Let us try to formalize the problems we have just considered from the point of view of set theory. What are we actually doing now? We numbered the hotel rooms. Such a situation in mathematics is called a countable set: an infinite set of K, which can be renumbered. An infinite set is called countable if there is a one-to-one correspondence between the elements of the set K and the natural numbers {1, 2, ...}. The formal presentation of the tasks we considered is as follows:
But let us move on to the sets that are more familiar to us — numerical. And let's start with natural numbers. Obviously, the set of natural numbers (ℕ) is countable simply by definition. If we add zero and negative numbers to natural numbers, we get a set of integers. Is this a countable set? We have already said above that if we combine two countable sets, the result will also be a countable set. From the addition of 0, the set will not lose the properties of being countable.
Next, consider rational numbers. This is a set of numbers that can be represented as m / n, where m is an integer and n is a natural number:
Thus, rational numbers are a countable number of countable sets and, when combined, will also become a countable set.
Algebraic numbers are the set of roots of polynomials with integer coefficients. Will this set be countable? The sets of roots of polynomials of different degrees are countable sets, which means that the result of their union will also be a countable set. From this it follows that the set of algebraic numbers (𝔸) is a countable set.
There are so-called transcendental numbers: real (ℝ), but not algebraic (𝔸) numbers are included in this set (𝕀). This set includes numbers e, π, etc.
Since we have mentioned the real numbers, we’ll dwell on them in more detail, especially since their set is not countable, which we will now prove. And for this you need to prove that the set of infinite decimal fractions is uncountable. For simplicity, let's imagine that after the comma we can have only zeros and ones:
Suppose that we have an infinite table in which all the numbers of the form we have indicated are numbered.
Let us prove that such a table cannot exist, we construct such a fraction, which is impossible to meet in this table:
0, x 1 , x 2 , x 3 , x 4 , xx 5 ...
Where x 1 ≠ a 1 , x 2 ≠ b 2 , x 3 c 3 , etc. This is called the Cantor diagonal process, it leads us to a number that is not in our table. So, we have a lot of uncountable.
After watching the lecture to the end, you will learn what are the problems associated with infinite sets, and how they are solved.
Gilbert hotel
Imagine a hypothetical hotel with an infinite number of rooms. All hotel rooms are already settled. but at some point another guest arrives at the hotel. Despite the fact that every hotel room already has at least one tenant, you can still find a room for a new one. After all, our hotel is infinite, and if you ask all the guests to move one room further, the very first room will be free. And even if several new guests arrive, by a series of similar relocations, it will be possible to find a place for them.
Now imagine that we have three such Hilbert hotels. Two of them are closed for repairs, and guests need to settle in one remaining. Even such an unusual case should not cause any problems, all guests will find a place in the third infinite hotel. For example, we can consistently resettle guests from two hotels on even and odd numbers, respectively.
')
Imagine an even more strange situation: an infinite number of Hilbert hotels are being closed for renovation, and we need to settle all their guests in one endless hotel. If we use the technique from the previous example, but we will settle the guests consistently diagonally, sooner or later we will be able to distribute all.
Let us try to formalize the problems we have just considered from the point of view of set theory. What are we actually doing now? We numbered the hotel rooms. Such a situation in mathematics is called a countable set: an infinite set of K, which can be renumbered. An infinite set is called countable if there is a one-to-one correspondence between the elements of the set K and the natural numbers {1, 2, ...}. The formal presentation of the tasks we considered is as follows:
- K is a countable set, therefore, K∪ {a} is also a countable set.
- K 1 , K 2 are countable sets, therefore K 1 ∪ K 2 will be a countable set.
- K 1 , K 2 , K 3 , K 4 , ... are countable sets, therefore, the combined set will also be countable:
Countable and uncountable sets
But let us move on to the sets that are more familiar to us — numerical. And let's start with natural numbers. Obviously, the set of natural numbers (ℕ) is countable simply by definition. If we add zero and negative numbers to natural numbers, we get a set of integers. Is this a countable set? We have already said above that if we combine two countable sets, the result will also be a countable set. From the addition of 0, the set will not lose the properties of being countable.
Next, consider rational numbers. This is a set of numbers that can be represented as m / n, where m is an integer and n is a natural number:
Thus, rational numbers are a countable number of countable sets and, when combined, will also become a countable set.
Algebraic numbers are the set of roots of polynomials with integer coefficients. Will this set be countable? The sets of roots of polynomials of different degrees are countable sets, which means that the result of their union will also be a countable set. From this it follows that the set of algebraic numbers (𝔸) is a countable set.
There are so-called transcendental numbers: real (ℝ), but not algebraic (𝔸) numbers are included in this set (𝕀). This set includes numbers e, π, etc.
Since we have mentioned the real numbers, we’ll dwell on them in more detail, especially since their set is not countable, which we will now prove. And for this you need to prove that the set of infinite decimal fractions is uncountable. For simplicity, let's imagine that after the comma we can have only zeros and ones:
Suppose that we have an infinite table in which all the numbers of the form we have indicated are numbered.
| 0, a 1 , a 2 , a 3 , a 4 , a 5 ... |
| 0, b 1 , b 2 , b 3 , b 4 , b 5 ... |
| 0, c 1 , c 2 , c 3 , c 4 , c 5 ... |
| 0 d 1 d 2 d 3 d 4 d 5 ... |
| ... |
| ... |
| ... |
Let us prove that such a table cannot exist, we construct such a fraction, which is impossible to meet in this table:
0, x 1 , x 2 , x 3 , x 4 , xx 5 ...
Where x 1 ≠ a 1 , x 2 ≠ b 2 , x 3 c 3 , etc. This is called the Cantor diagonal process, it leads us to a number that is not in our table. So, we have a lot of uncountable.
After watching the lecture to the end, you will learn what are the problems associated with infinite sets, and how they are solved.
Source: https://habr.com/ru/post/216665/
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