How big is infinity?

Did you know that some infinities are bigger than others? Or are we not exactly sure if there are infinities between two infinities? Mathematicians have been thinking about the second question for centuries, and it's interesting to know that some of the work done on infinities has changed the way mathematicians think about this! How big is infinity?

Did you know that some infinities are bigger than others? Or are we not exactly sure if there are infinities between two infinities? Mathematicians have been thinking about the second question for centuries, and it's interesting to know that some of the work done on infinities has changed the way mathematicians think about this!

How do they measure infinities!

To answer questions about understanding the size of infinite sets, let's start with sets of numbers that are easier to count. Consider a set of objects, numbers, or elements that contain a finite number of these elements. How big is infinity?

Here we consider two examples of four-membered and finite sets.

As you can see, determining the size of a finite set is easy, because you only need to count the number of elements in it. Since the set is finite, you know that eventually you will stop counting and you will get the size of the set. At the same time, it is clear that this strategy is not efficient for infinite sets. Therefore, we proceed in a different way. Here we consider the set of natural numbers denoted by . Of course, you may argue that zero is not a natural number, but this argument does not affect our research on infinities.

={0,1,2, 3,4,5,}

What is the size of this set?

Since there is no largest natural number, your attempt to count the number of elements in this set is pointless. In this regard, one solution is to simply declare the size of this infinite set to be infinite, which is not wrong, but when you deal with more infinite sets, you will realize that this is not so true!

Consider a set of real numbers that all numbers can be expressed in a decimal expansion 7, 3.2-8.015 or an infinite expansion like 2-1.414213=1.414213. Since every natural number is also a real number, we can say that the set of real numbers is at least as large as the set of natural numbers, so it must also be infinite. But the expression of the infinite size of the set of real numbers is equally unacceptable. In order to understand this issue, we choose two numbers like 3 and 7. It is natural that between these two numbers there is a finite number of natural numbers such as 4, 5, 6, but the real numbers between these two numbers are infinite (, 4.01023, 5.666, etc.).

So no matter how close two distinct real numbers are to each other, there will always be an infinite number of real numbers between them. Of course, you should know that this issue in itself does not mean that the sets of real numbers and natural numbers have different sizes, but it shows that there is something fundamentally different about these two infinite sets that requires further investigation! In this regard, Georg Cantor investigated this issue at the end of the 19th century. He showed that these two infinite sets have different sizes in the real sense. To understand this, you must first learn how to compare infinite sets, which is possible with the help of functions. , parabolic plots on the Cartesian plane, commands such as take the input and add 3 to it, etc. In this section, we use a function to match the elements of one set to the elements of another set. Considering the same problem, for the first set we consider natural numbers () and for the other set, which we call S, we choose all even natural numbers.

N={0,1,2,3,4,...} S={0,2,4,6,8,...}

Simple function f(x)=2x elements of converts to S elements. This means that this function doubles its inputs, so if we consider the elements of as the input of f(x) (the set of inputs of a function is called its domain!), the outputs will always be the elements of S. For example f(0)=0, f(3)=6 and so on! You can do this by lining up the elements of the two sets next to each other and then using arrows to show how the function f transforms the inputs from to the outputs in S. How big is infinity?

Notice how f(x) assigns exactly one element of S to each element in . First, f assigns all the elements in S to . In more technical language, every element of S is the image of an element of in the function f, for example, the even number 3472 in S is written as f(x)=3472, and its counterpart to is the number 1736 . In other words, the function f(x) maps the numbers onto S. In general, the first important point in converting the elements of two sets to each other is that the function f(x) (a "surjective" spanning function) transforms the inputs from the set into an output in S and Nothing is lost in the set S.

The second peculiarity about how the outputs of f(x) are assigned to the inputs is that no two elements in become the same element in S. This means that if two numbers are different in , twice their number will also be different. In this case, we say that our function is one-to-one "injective" (also written as 1-1!). Therefore, we emphasize that each element in S is paired with only one element in .

These two properties of the function f(x) combine in a powerful way, which means that the function f(x) matches Creates a completeness between the elements of and the elements of S! The fact that f(x) is a spanning function means that everything in S has a partner in , and the fact that the function is 1-to-1 means that nothing in S becomes two partners in . In short, the function f(x) pairs each element of with exactly one element of S.

Functions that are both spanned and one-to-one are called bijections. And since such a function together with these two properties create a 1 to 1 correspondence between the two sets, it means that every element in one set has exactly one partner in the other set, and this is one way to show the same size of two infinite sets.

Since every element in one set has exactly one partner in another set, one way to show that two infinite sets have the same size is

The one-to-one and spanning functions create a 1-to-1 correspondence between two sets, meaning that each element in one set has exactly one partner in the other. Such a method is a convenient approach to show that two infinite sets have the same size.

Since our function f(x) is a one-to-one spanning function, it shows that two infinite sets and S are of the same size. . Now it may be asked that isn't every even natural number itself a natural number and includes everything in S along with its members, so shouldn't be bigger than S?

In answer, it should be said that If we were dealing with finite sets, you would be right. But in the case of an infinite set, it can be stated that although a kind of infinite set can contain another set, it can still be the same size as that set. In other words, the size of the infinite set plus 1 is equal to infinity; Of course, this is only one of the amazing features of such collections! Which is another amazing property of other infinities is that there are infinite sets of different sizes! Of course, it should be mentioned that we mentioned this topic before and as you have seen, infinite sets of real and natural numbers have different sizes according to the contour proof.

Comparison between infinities

Since Since there is an infinity of real numbers between both real and distinct, let us focus in this section on the infinity of real numbers between zero and 1. As you know, each of these numbers can be considered as a (possibly infinite) decimal expansion as shown below: How big is infinity?

Here a1,a2,a3 and the rest are known as digits and all of them should not be zero so that the number zero is not displayed. Now, according to the diagonal argument, what happens if there is a one-to-one spanning function between the natural numbers and these real numbers? In such a case, the size of the two sets will be the same, and this function can be used to match any real number between zero and one to a natural number. In this case, a sorted list of matches will be created as follows: How big is infinity?

Interesting part The diagonal proof is that this list can be used to construct a real number that is not in the list. For this purpose, the following procedure can be followed: Choose the first digit after the decimal point different from a1, change the second digit from b2, change the third digit from c3 and continue in the same way. How big is infinity?

In this case, the real number is defined by the relationship with the diagonal numbers of the list. Naturally, this real number is not the first number in the list because the first number in the list is different. It also cannot be the second number in the list, because the second digit is also different. This number cannot be the nth number in this list, because it has a different nth digit and this is true for all ns in the list, so this new number, which is between zero and 1, does not exist in the list.

On the other hand It was assumed that all real numbers between 0 and 1 are in the list, which was clearly violated. This contradiction comes from the assumption that there is a one-to-one spanning function between natural numbers and real numbers between zero and 1, but the assumption was violated and it was found that such a function cannot exist in the list. Therefore, these infinite sets have different sizes. By fiddling with functions a bit, it can be shown that the set of all real numbers is equal to the set of all real numbers between zero and one, so the real numbers that include the natural numbers They can be an infinite and larger set. It should be noted that the technical term for the size of an infinite set is something called "cardinality". provides their members.

During these investigations, the diagonal proof showed that the cardinality of real numbers is greater than the cardinality of natural numbers. The cardinality of natural numbers is written with 0, which is pronounced "aleph naught"; In a standard view this is the smallest cardinal of infinity. The next infinite cardinal is 1 (alpha one), and following that was a simple question that has puzzled mathematicians for more than a century: is 1 a real numerical cardinality? In other words, are there other infinities between natural numbers and real numbers? Cantor thought the answer was no, but he was unable to prove it. Even in the early 1900s, this question was considered so important that when David Hilbert compiled his famous list of the 23 most important and unanswered problems in mathematics, it was his number one problem. Until 1940, a famous logician named Kurt Gdel proved that it is impossible to prove the existence of infinities between natural and real numbers based on the general rules accepted in set theory. At the time, this was a big step towards proving Cantor's hypothesis correct, but two decades later, a mathematician named Paul Cohen showed that it is impossible to prove that such an infinity does not exist!

So it seems that mathematics With its abstract, scary and sweet world, it still interests mathematicians.

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