Different Types of Infinities (Wherein I will try to convince you that there are different types of infinities, and that some are ‘bigger’ or ‘more abundant’ than others.)
I’m going to concentrate on showing that integers and real numbers fall into two different categories of infinity. This is an informal proof, meaning that it lacks the rigor that would qualify as a proof for a mathematician, but should (hopefully) be convincing to a layperson. If you are interested in delving deeper, I recommend starting with this video by Vi Hart.
The guy who did the breakthrough work in the mathematics of infinity was Georg Cantor. Before Cantor came along infinity was taken to be a single concept of numbers going on forever, but Cantor showed that the picture was much more complicated and weird. In fact, some of the leading mathematicians of Cantor’s day rejected Cantor’s work and pilloried him.
First, integers can be seen as a special case of real numbers that have all zeros after the decimal. We typically ignore the decimal when writing down integers, but another way to write the number 5 is 5.00000000000000… and 3843974937 is 3843974937.00000000000000… (the ellipses “…” denote that the zeros repeat forever). You can see in this diagram that the integers are a subset of the rationals which are in turn a subset of the reals. So, although both sets are infinite, one is a subset of the other.
Now, let’s look at a line segment (i.e. a line of finite length) on the number line, even the smallest line segment we can imagine, will encompass an infinity of real numbers, but only 0 or a finite number of integers.
For example, this line segment from 3.10… to 3.20… contains 0 integers, but an infinity of real numbers, including the irrational number, π (3.1415926…).
This line segment, from 1.0… to 1,000,000.0… contains 1,000,000 integers, and an infinity of real numbers.
Cantor developed several proofs, including the diagonal argument, showing the one to one relationship between integers and rationals, and also how the reals cannot be put into a one to one relationship with the integers. There are plenty of websites and videos that explain this concept. So, I’m going to present my own ‘cinematic proof’:
Imagine your are in a desert that stretches to the horizon in all directions. Each grain of sand represents a real number, and you are searching for the sand grains that are integers. You are about to pass out from the heat when you see something in the distance. It starts as a dot wavering in the heat waves but eventually you can make out that it’s a person on camelback. Eventually, Omar Sharif is standing before you. He presents you with an easy to follow map to the location of all the integers in the desert, and a scoop that can be adjusted in size to scoop as many of the integers as you want. When you scoop the sand and sift out the integers, you wind up with a small pile, and the sand you discard is a dune that piles up to the sky. That is how the integers and real numbers relate to each other.