Shulou(Shulou.com)11/24 Report--

David Hilbert (Picture Source: Wikimedia Commons) imagine a hotel with an infinite number of rooms.

When you arrive at the hotel, the room is full (maybe his family has a large guest list), and you need the receptionist to find a room for you.

The receptionist was smart enough to move guests in room 1 of the hotel into room 2, while guests in room 2 moved to room 3-that is, guests in room n moved into room n + 1. You can imagine the scene when they moved together.

When everyone was settled, Room 1 was vacant and all the tenants were well arranged.

Well, this example shows ∞ + 1 = ∞.

In a hotel with unlimited rooms but full rooms, you can always find one more room.

David Hilbert, the genius David Hilbert, the greatest and most prolific mathematician of the 20th century, introduced this example to explain the most counterintuitive concepts of infinite set and transfinite arithmetic (transfinite arithmetic) in his lecture "beyond ber das Unendliche".

Hilbert hopes to demonstrate the concept of "infinity" and a reasonable way to use it to calculate. He said, "Don't be afraid of it," but accept it systematically and in all directions.

Georg infinity (∞) had been considered "taboo" in mathematics, and the first mathematicians who tried to build the concept on a solid foundation, such as Georg Georg Cantor, were severely criticized, but Hilbert still found charm in the theory.

Some of the problems that arise from the counterintuitive infinite set (infinite sets) are expressed through Wikipedia:

In the case of an infinite number of rooms, the expressions "there is a guest in each room" and "no more guests can be accommodated" are not equivalent.

Infinity is not a real number or a complex number, so it is important to pay special attention to the expression ∞ + 1 = ∞. This equation does not make any sense before we define the "number space" of calculation.

But of course we can assume that there is such an extension space so that we can add and subtract ∞.

Going back to this thought experiment, what should we do when k guests arrive at this full hotel and we need another k rooms?

Again, we just need to let the guests in each room move into the room with the current room number + k, so that the first k rooms are available and available. For example, if three people come to the hotel and each needs a room, then the guest in room 1 needs to move into room 4, the guest in room 2 needs to move to room 5, and so on.

But how to understand ∞-1?

At the Infinity Hotel, we just need to let the guest in room 1 leave the hotel, and then move the guest in room 2 to room 1, and the guest in room 3 to room 2, so we get another ∞:

∞-1 = ∞.

There is no problem at present to reach infinity.

But what if there are an infinite number of guests? The infinity mentioned here is countable infinity, which mathematically means that they can be marked one by one with positive integers.

This time, we will not be able to make an infinite move for all the guests as before. But we can move the guests in room 1 to room 2 and the guests in room 2 to room 4, that is to say, the guests in room n move to room 2n.

This process does not need to take into account the difficulty of moving to a larger room for guests living in a large room-we just need to know that it is mathematically feasible.

Now all the odd-numbered rooms are available to accommodate all the new guests.

The above process is actually saying that the "scale" of this infinite set of natural numbers, called cardinality (cardinality, or "potential"), is the same as the cardinality of the set of even natural numbers (although it is a subset of the set of natural numbers).

Wait a minute... Does this mean that the numbers of natural and even natural numbers are the same?

Well, actually, it's true...

In mathematics, in this case, the function f (n) = 2n is defined to represent the corresponding relationship between even natural numbers and natural numbers to compare two infinite sets. This function has an inverse function, g (n) = n / 2, which represents the corresponding relationship from even natural numbers to natural numbers.

In addition, f has the property of bijective, which means that all even natural numbers are the values of f with natural numbers as independent variables, and every even natural number has and at most one such independent variable. Such a correspondence is called "one-to-one mapping", and it is also surjective, in which the function mappings in two sets can be matched by elements.

Imagine what to do if you can't count but need to compare two piles of stones which one is more?

At this point, you can pair two piles of stones until there is only one pile of stones or none left, and then stop combining. If there is only one pile left, then this pile of stones has a larger initial number; if the two piles can be paired and eventually match exactly, it means that the initial number of the two piles of stones is the same.

We use this concept to compare two infinite sets, which are exactly the same as the stone scene above. It's just that pairing occurs not between stones but between bijective functions.

A bigger infinity. What if there are an infinite number of buses that can carry an infinite number of guests, and we need to book a separate room for all the guests?

No problem!

First, each bus and each seat on it is marked with a natural number (after all, within a countable infinite set), and each corresponding person has a unique "address" consisting of two numbers: one number s represents the seat number on the bus, the other number b represents the bus number; make the bus number of the hotel guest b = 0.

Just put everyone in a room with room number 2s 3b.

For example, the guest with room number 1119744 = 2937 corresponds to seat 9 on bus No. 7. This case can be extended to more layers by using primes.

For example, if there are an infinite number of ferries, each carrying an infinite number of buses and each bus with an infinite number of passengers, then we can still expand the hotel to arrange accommodation as above, only need to introduce the prime number 5 into the corresponding operation.

The existing three-layer infinity nesting or more finite nesting can be solved in this way-after all, there are infinitely many primes, and we can't actually run out of them.

Prime decomposition is only one of many ways to solve this problem, but at least it can be solved. The problem is what if there are infinite layers of nesting? This kind of problem may not be solved.

This is because some infinity is bigger!

Yes, when the problem of "scale" or cardinality of a set is introduced, infinity is not just infinity. For example, the set of all fractions (the numerator denominator is an integer) is the same as the cardinality of the set of natural numbers.

Yes, the number of scores is the same as that of natural numbers, although natural numbers can also be expressed as fractions.

Readers can prove it by looking for the bijection between two sets, that is, pairing each element in the two sets to find an one-to-one mapping relationship.

If there is bijection between a natural number set and a given set A, then the set An is countable, and the cardinality of An is the natural cardinality, denoted as ℕ, and sometimes as ℵ0.

Continuum, so there is some infinity to be "bigger", so there are some interesting questions left behind--

What are the sets that are larger than the natural cardinality? What does an uncountable collection look like? How many different infinities are there?

A good example of a set that is larger than the cardinality of natural numbers is the set of real numbers recorded as ℝ. This set is uncountable, which means that there is no bijective correspondence between ℝ and ℕ.

The set of real numbers includes not only all fractions, but also numbers such as pi and e that cannot be written as fractions made up of integers.

The problem of studying whether there is other infinity between the cardinality of natural numbers and real numbers is called the continuum hypothesis (continuum hypothesis), but in our existing axiomatic system, this hypothesis can not be proved to be true or false.

Such marginal philosophical problems involve meta-mathematics and mathematical logic, in which we can discuss whether we need to find an alternative axiom system to solve this problem, and then create a completely different mathematical system. If you go any further... All I can say is that it was fun!

In addition, there are many kinds of infinity, so which one are we talking about?

Original text link:

Hilbert's Hotel-An IngeniousExplanation of Infinity

Some of the pictures in this article come from the Internet.

The translated content only represents the author's point of view, not the position of the Institute of Physics of the Chinese Academy of Sciences.

This article comes from the official account of Wechat: Institute of Physics, Chinese Academy of Sciences (ID:cas-iop), author: Kasper M ü ller, translator: zhenni, revision: Dannis

Welcome to subscribe "Shulou Technology Information " to get latest news, interesting things and hot topics in the IT industry, and controls the hottest and latest Internet news, technology news and IT industry trends.