Why do some infinities seem bigger than others, and how do mathematicians compare their sizes?

Question

Why do some infinities seem bigger than others, and how do mathematicians compare their sizes?

In: Mathematics

Answer 1

In terms of sets we compare sizes by whether or not we can draw a one-on-one correspondence between the elements of those sets. If we can do that then we consider those sets to be the same size. This is fairly simple when it comes to sets that have a finite number of elements but can get a little weird when we have sets with an infinite number of elements.

As it turns out, there are sets with infinite numbers of elements that we can’t draw one-to-one correspondences between, which means that one is bigger than the other, despite the fact that both are infinite in size. An example of this is the integers and the real numbers.