There are two interesting kinds of infinity. The first is the infinite number of Counting Numbers corresponding to unbounded progressions. The second is the infinitely dense number of Real Numbers corresponding to a continuous field of points in which every pair of points have an infinite number of points between them.

Almost all understanding starts with knowing How Many;or maybe it sounds better negatively, I know very little about anything if I can't answer the question How Many there are of it. The obvious way to answer the question of how many is with a counting number: 0, 1, 2, 3, ... 10034, 10035, ...

I discovered the shortcomings of counting numbers early in life in the same way many of us did. For example:
You: I hate you.
Me: Well, I hate you twice as much
You: I hate you three times as much
Me: I hate you a thousand times as much
You: I hate you a million times as much
Me: I hate you a million billion zillion times as much
You: I hate you INFINITY times a much
Me: (pause) I hate you two times infinity as much

This is the first of the InfiniteVarieties; the number of coiunting numbers is unbounded. The number of counting numbers is frequently designated as <img src="" /> (aleph0).

An interesting question is: is two times infinity more than just one infinity? To answer this, we need to examine how we can decide that one set is bigger or smaller than another particularly when the sets in question may be uncountably large.

Suppose I had a bunch of apples (A red one, a green one, a yellow one and one with the leaf still on) and a group of people (Steve, Kris and Kristin) and I wanted to say whether I had more people than apples, more apples than people or the same number of apples as people. But let's suppose that I couldn't count the things (maybe because one of the sets was infinite).

First of all, I could assign Steve to the red apple, Kris to the green one and Kristin to the yellow. On the basis of this assignment I know that there are at least as many apples as people because each person can be assigned aunique apple.

Second, I could assign the yellow apple to Kris,the red apple to Kristin and the one with a leaf to Steve. When I look for a person to assign the green apple to, there are no unused people. So I know that there are fewer people than apples.

Counting is certainly more efficient but it is equivalent to this process with the standard sets of three and four things being used when I count respectively the three people and four apples.

The process seems silly when dealing with small finite sets; it is less so when we are comparing Infinite Sets. The most common technique for comparing the size of infinite sets is based on the Cantor-Bernstein Theorem and mirrors the apples to people comparison above.

If we consider the set of counting numbers (N) being 1,2,3,4... and the set of even number (EN) 2,4,6,8,... we encounter one of the central properties of infinite set. Intuitively, since the set of even numbers is a proper subset of the counting numbers, the counting numbers should be larger than the even numbers.

But let's apply the apples to people method. Clearly there is an unique assignment within the counting numbers for each of the even numbers (since the evens are a proper subset of the counting numbers). But we can also pair each counting number (i) with a unique even number (2i). This establishes the correspondence between the two sets and we are let to conclude that the size of the set of counting numbers is the same as the set of even numbers.

Getting back to my attempt at one-upmanship. I assumed that doubling up on infinite would produce a larger set. Unfortunately, and counter-intuitively, adding infinites together does not produce a larger set. In particular, twice infinity is not larger than the set of counting numbers. This is a profoundly counterintuitive result and I think it is worthwhile to work through the details.

What did I mean by "two times infinity?" Since my basic infinity is the set of counting numbers, to get to twice that we need another set the same size. I'm going to imagine that ordinary counting numbers are black and that I have an extra set all decked out in red. So twice infinity is the black numbers plus the red set.
To compare size of the plain old infinite set of counting numbers with the bi-colored twice infinity set we will use a mapping strategy (i.e. the Cantor/Bernstein Theorem). From the natural numbers to the bi-colored set is easy; just map each natural number to the corresponding black number. From the bi-colored set to the natural numbers, we will use the following rule:
if a number is black, map it to twice its value
if a number is red, map it to twice its value plus one.
This map is 1-1 from the bi-colored set into the natural numbers for each value of x 2*x is different than anyother value, as is 2*x+1. The blacks map to even numbers and the red to odds so they don't overlap. Yes, I left the natural number 1 out of the mapping, but Cantor/Bernstein saves us from needing to be so fuzzy. It is enough that each bi-color number is mapped to a unique natural number. And the "obvious mapping" maps each natural number into a unique bi-color number.

This demonstrates that "two times infinity" is no larger than the infinity of counting numbers. But it is worthwhile looking for an intuition that will help us internalize this weird result. For me, it helps to think of infinity of counting numbers, as going on forever. It makes sense to me, that going on forever and then going on some more is still going on forever.