For most people, "the whole is greater than the part" is an undeniable and self-evident truth. The Greek mathematician Euclid even wrote it as an axiom in his work Elements of Geometry. Now, if someone told you that the whole and its parts could possibly have the same number, would you be surprised?
To explain this issue, let's first consider what it means to say that two things have the same number. For example, if we see a packed audience at a performance with no standing room, we can deduce that the number of seats and the number of people are the same without having to count them. This is because there is a one-to-one correspondence between the seats and the people. In abstract terms, if two sets can be put into a one-to-one correspondence, then their elements are equal in number. Obviously, it's impossible to establish a one-to-one correspondence between a finite set and its part. But what about infinite sets?
In 1638, Italian physicist Galileo recorded the following surprising fact in his book: every positive integer corresponds to its square, so there are as many squares as positive integers. But squares are only a part of positive integers, so positive integers can have the same number as their parts. Galileo's discovery did not attract much attention at the time. Two hundred years later, a Czech mathematician named Bolzano reiterated the same fact in a more general form: an infinite set can be put into a one-to-one correspondence with its proper subset. However, this still did not gain widespread attention. It wasn't until 1872 that German mathematician Cantor began conducting in-depth research on infinite sets. Cantor was convinced that the fact that a set can be equal in size to its proper subset is the essential characteristic of infinite sets. Moreover, he discovered that there are many more unexpected phenomena related to infinite sets.