One way to see that there does in fact exist an object that can serve as a measure of the size of a set is just to use the collection of all sets of that size as your object. This is an interesting topic, so we will discuss it briefly.
Many infinite sets have the same size.
For instance, the set of even integers has the same size as the set of all integers. In one sense that may seem surprising, since the second set has all the elements of the first set in it plus all the odd integers as well, but it's fact of life about infinite sets that you can add elements to an infinite set without altering its size. For instance, the set of all real numbers is a much bigger set. These concepts that describe the sizes of infinite sets are called infinite cardinal numbers .