The founder of set theory, German mathematician Cantor, proposed a theory of infinite sets that was considered revolutionary at the time of its inception. Initially, his theory faced significant resistance and even public opposition. However, many mathematicians, including the great 20th-century mathematician Hilbert, strongly supported this theory. It is said that Hilbert once used the following story to illustrate the properties of infinity during a lecture.
Imagine a hotel with a limited number of rooms, all of which are already occupied. A new guest arrives and requests a room. The hotel owner says, "Sorry, all the rooms are full." Now imagine a hotel with an infinite number of rooms, all of which are also fully booked. A new guest arrives and requests a room. "But... well, no problem!" the hotel owner says after thinking for a moment. Next, he moves the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, and so on, until the new guest can move into the vacated Room 1.
Still in this hotel, and all rooms are fully booked. An infinite number of new guests arrive and request rooms. Based on his previous experience, the hotel owner promptly says, "Okay, gentlemen, please wait a moment." He then moves the guest in Room 1 to Room 2, the guest in Room 2 to Room 4, and the guest in Room 3 to Room 6, and so on. In this way, all the original guests move into even-numbered rooms, and the infinite number of new guests can directly move into the vacated odd-numbered rooms. Thus, all the new guests are satisfactorily accommodated.
That's the story of this anecdote. Later, this hotel became known as the Hilbert Hotel. Hilbert told this story to help people better understand the concept of "infinite sets". The reason why the hotel owner could always solve the problem of accommodating new guests in a fully booked situation was because he established a one-to-one correspondence between an infinite set and another infinite set each time.