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At the end of the Marvel blockbuster Avengers: the final Battle, Tony Stark said goodbye to his young daughter through a pre-recorded hologram: "I love you 3000 times." This touching moment echoes a previous scene in which the two were performing interesting bedtime rituals to quantify their love for each other. Stark is played by Robert Downey Jr. (Robert Downey Jr.) He said the line was inspired by similar communication between him and his children. This game is an interesting way to explore big numbers:

"I love you ten times."

"but I love you a hundred times."

"well, I love you 101 times!"

This is why "googolplex" has become a buzzword. But we all know the end result of this debate:

"I love you infinitely!"

"Oh, really? I love you infinitely plus one!"

Whether playing on the playground or in bedtime communication, children are exposed to the concept of infinity long before math class and are naturally infatuated with this mysterious, complex and important concept. Some children grow up to be mathematicians fascinated by infinity, while others discover novel things about infinity.

You may know that some sets of numbers are infinite, but do you know that some infinities are larger than others? And do you know that we are not sure whether there is any other infinity between the two most familiar infinities? Mathematicians have been thinking about the second problem for at least a century, and some recent work has changed people's view of it.

To solve the problem of infinite set size, let's start with sets that are easier to count. A collection is a collection of objects or elements, and a finite set is a collection that contains a finite number of objects.

For two examples of finite sets, each set has four elements. It is easy to determine the size of a finite set: just count the number of elements it contains. Because the set is limited, you must be able to count it, and eventually you will know the size of the set.

This strategy does not apply to infinite sets. Here is a set of natural numbers, expressed as ℕ. Some people may say that 0 is not a natural number, but this argument does not affect our study of infinity.)

How big is this collection? It is not feasible to try to calculate the number of elements because there is no maximum natural number. One solution is to simply declare the size of this infinite set as "infinite", which is not wrong, but when you start to explore other infinite sets, you will realize that this is not entirely true.

Consider a set of real numbers that can be represented by decimal (finite) expansions, such as 7 − 3.2, or infinite expansions. For example, because every natural number is a real number, the set of real numbers is at least as large as the set of natural numbers, and therefore must be infinite.

However, it would be imperfect to think that the size of a set of real numbers is the same as that of a set of natural numbers. To prove this, choose two numbers randomly, such as 3 and 7. There is always a limited number of natural numbers between these two numbers: here are 4, 5 and 6. But there are always an infinite number of real numbers between them, such as 3.001, 3.01, π, 4.01023, 5.666, and so on.

It is worth noting that no matter how close any two different real numbers are to each other, there is always an infinite number of real numbers in the middle. Although this fact does not mean that the size of the real number set is different from that of the natural number set, it does show that there are some essential differences between the two infinite sets worthy of further study.

Mathematician Georg Cantor studied this at the end of the 19th century. He proved that the sizes of the two infinite sets are indeed different. To understand and appreciate how he did it, we must first understand how to compare infinite sets. The "secret means" used in it is the main content of all math classes: functions.

There are many different ways to represent a function-a function expression in shape, a parabolic graph on a Cartesian coordinate plane, or a function correspondence such as "square the input and add 1". But here we think of a function as a way to match the elements of one set with those of another.

Suppose one of the sets is ℕ, that is, the set of natural numbers. We select all the even numbers to form another set and call it S. Here are our two groups:

There is a simple function that converts the element of ℕ into the element of S: F (x) = 2x. This function just doubles its input, so if we think of the element of ℕ as the input of f (x) (we call the input set of the function "domain"), the output is always the element of S. For example, f (0) = 0 ~ (1) = 2 ~ ~ f (2) = 4 ~ ~ f (3) = 6 and so on.

You can visualize this: put the elements of the two collections side by side and use arrows to indicate how the function f converts the input of ℕ into the output of S.

Notice how f (x) maps an element in S to each element in ℕ. That's what functions do, but what f (x) does here is special. First, f allocates everything in S to the corresponding element in ℕ. In functional terms, we say that every element of S is the "image" of an element of N under the function f. For example, if there is an even number 3472 in S, we can find an x in ℕ so that f (x) = 3472. In this case, we say that the function f (x) maps ℕ to S. A more gaudy statement is that the function f (x) is "surjective". No matter how it is described, it is important that when the function f (x) converts the input of ℕ to the output of S, nothing in S is lost in the process.

The second special thing about how f (x) allocates output to input is that the two elements in ℕ are not converted to the same elements in S. 5 and 11 are different natural numbers in ℕ, and their output in S is also different: 10 and 22. In this case, we say that f (x) is "1 to 1" (also written as "1-1"). We describe f (x) as "single shot". The key here is that nothing in S is reused: each element in S is paired with only one element in ℕ.

These two features of f (x) are combined in a powerful way. The function f (x) creates a perfect match between the elements of ℕ and the elements of S. F (x) is "surjective", which means that all elements in S have a partner in ℕ, while f (x) is "1-to-1", which means that any element in S has no two partners in ℕ. In short, the function f (x) matches each element of ℕ with a unique element in S.

A function that is both unijective and surjective is called bijection, which creates a 1-to-1 correspondence between two sets. This means that each element in one set happens to have a corresponding element in another set, which is a way to indicate that two infinite sets have the same size.

Since our function f (x) is a bijection, it shows that the two infinite sets ℕ and S are of the same size. This seems surprising: after all, every even natural number is itself a natural number, so ℕ contains S and more. Shouldn't ℕ be bigger than S? If we are dealing with a finite set, the answer is yes. But an infinite set can completely contain another infinite set, and at the same time they are of the same size. Just as "infinity plus one" is no more loving than ordinary "infinity", this is just one of the many amazing properties of the infinite set.

What is even more surprising is that there are countless sets of different sizes. Earlier, we discussed the different properties of infinite sets of real numbers and natural numbers, and Cantor proved that the two infinite sets have different sizes. He did this with an ingenious and famous diagonal argument.

Because there are infinitely many real numbers between any two different real numbers, we now only focus on the infinite number of real numbers between 0 and 1. Each of these numbers can be thought of as a (possibly infinite) decimal expansion, like this.

Here, and so on are just numeric symbols, but we require that not all numbers are zeros, so we do not include the number 0 itself in the set.

The diagonal argument starts with a question: what happens if there is a bijection between natural numbers and real numbers? If there is such a bijection, then the two sets will be the same size, and you can use this function to match each real number between 0 and 1 with a natural number. You can imagine an ordered list of matches, like this:

The genius of diagonal argument is that you can use this list to construct a real number that cannot be on the list. Construct a real number bit by bit as follows: the first number after the decimal point is different, the second number is different, the third number is different, and so on.

This real number is defined by its relationship to the diagonal of the list. So is it on the list? First of all, it can't be the first number in the list, because its first number is different. And it can't be the second number in the sequence, because its second number is different. In fact, it cannot be the nth number in this series because its nth digit is different. This is true for all n, so this new number between 0 and 1 cannot be in the list.

But all real numbers between 0 and 1 should be on the list! This contradiction arises from the assumption that there is a bijection between the natural number and the real number between 0 and 1. Therefore, there is no such bijection. This means that the two infinite sets have different sizes. By doing a little more work with the function, it can be proved that the set of all real numbers is the same size as the set of all real numbers between 0 and 1, so the real number containing natural numbers must be a larger infinite set.

The technical term for the size of an infinite set is its "cardinality". The diagonal parameter indicates that the cardinality of the real number is greater than that of the natural number. The cardinality of a natural number is written as ℵ0 and read as "aleph 0". From a mathematical standard point of view, this is the smallest infinite cardinality.

The next infinite cardinality is ℵ1 ("aleph 1"), and a simple question has perplexed mathematicians for more than a century: is ℵ1 the cardinality of real numbers? In other words, is there any other infinity between natural numbers and real numbers? Cantor thought the answer was no-a claim later known as the continuum hypothesis-but he could not prove it. At the beginning of the 20th century, the problem was considered so important that when David David Hilbert made a list of 23 important mathematical open problems, he placed the continuum hypothesis at the top of the list.

A hundred years later, mankind has made great progress, but this progress has also brought new mysteries. In 1940, the famous logician Kurt Godel proved that under the generally accepted rules of set theory, it was impossible to prove that there was infinity between natural numbers and real numbers. This seems to be a big step in proving the continuum hypothesis. But 20 years later, mathematician Paul Cohen proved that it is impossible to prove that such infinity does not exist! It turns out that the continuum hypothesis cannot be proved in any case.

These results jointly establish the "independence" of the continuum hypothesis. This means that the generally accepted set rules do not fully explain whether there is infinity between natural numbers and real numbers. But instead of discouraging mathematicians in their pursuit of understanding infinity, it leads them in a new direction. Mathematicians are now looking for new basic rules for infinite sets that can both explain what is known about infinity and help fill in gaps.

Saying "my love for you is independent of axiom" may not be as interesting as saying "I love you infinitely plus one", but it may help the next generation of mathematicians who love infinity get a good night's sleep.

Exercise 1. Let T = {1, 3, 5, 7, … }, that is, a set of positive odd natural numbers. Is the number of elements of T greater than, less than, or as large as the natural number set N?

Answer: the same size. You can use the function f (x) = 2x+1 to map elements in N to corresponding elements in T in a way that has both surjective (onto) and monojective (1-1). This function is a bijection between N and T, and because there is bijection, the size of the set is the same.

two。 Find out the natural number set ℕ and the integer set Z = {... , − 3, − 2, − 1pm 0pm 1pm 2pm 3, … The 1-to-1 correspondence between}.

Answer: one way is to visualize the pairing list, like this:

You can also try to define a function that matches the element. Function

Mapping ℕ to Z is an one-to-one correspondence. So there are as many integers as natural numbers, which is another magical conclusion about infinity.

Author: Patrick Honner

Translation: C é cor

Revision: cloud opening and leaves falling

Original link: how big is infinity

This article comes from the official account of Wechat: Institute of Physics, Chinese Academy of Sciences (ID:cas-iop), author: Patrick Honner

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