Suppose you have two collections of objects. Call them collection A and collection B. How could you tell which is bigger, or if the two are the same size? Of course, you could just count all the objects in collection A, then count all the objects in collection B, and compare the two numbers. But it might be easier (and eliminate the risk of losing count), to try to match the two sets up: pair every object from A with one from B, until one or the other runs out. Sets which can be matched up are the same size, and sets which can’t are different. This idea could hardly be simpler, but in Cantor’s hands it yielded an extraordinary discovery: he proved that some infinite sets can never be matched with others. So immediately we have to conclude that there are different levels of infinity, with some bigger than others.
Da: Cantor and Cohen: Infinite investigators part I | plus.maths.org.
