How can infinity have different sizes? It’s just infinite, right?

Well, not so much. You see, from a mathematical standpoint, the term “infinity” is defined as being the size of the set of integers. So is it possible to have something bigger than this? Yepers, and it is best illustrated by trying to map one thing to another.

By “mapping” you can think of placing marbles into an egg carton, one per slot. If you’ve marbles left over, then the number of marbles is larger than the number of slots. So let’s start off with an infinitely long egg carton, each slot numbered 1, 2, 3, etc. Now well take an infinite number of marbles, with the same numbers on them. None left over, obviously.

Suppose we’ve twice as many marbles, this time labeled 1, 2, 3, etc.; and then -1, -2, -3, etc. We’ll even throw in an extra marble labeled zero. Is this the same size? Yes it is, because you can alternate marbles. First put the zeroth marble into slot one, then the number one marble into slot two, then the negative one marble into three, the two marble into four, the negative two marble into five, and so on. You never run out of marbles, of course, but you never run out of slots either. And the important point: for any specific marble, you know exactly which numbered slot it goes into. So the set of all integers (positive, negative and zero) is the same size as the set of all positive integers.

Let’s throw a curve: all rational numbers. For those of you whose math escapes you, a rational number can be expressed as the ratio of two integers. In other words, any particular rational number x can be expressed as ( a / b ) in which a and b are both (non-zero) integers. So how big is the set of rational numbers? It turns out that it’s still the same size of infinity. Here’s how:

Let’s make a table of all possible a’s and b’s:

egg_crate_empty

The grid is infinite in both the a and b directions as it contains all positive and negative integers along those axes.  So how do we map this onto the line of positive integers?  Just start in the “middle” (pick a point, any point) and spiral out.  Yep, it sounds like cheating, but again, you can compute where any marble on the a-b grid would be placed into the infinite egg carton.

egg_crate_full

So again, the set of all rational numbers is the same size as the set of all positive integers — regular ol’ infinity.  So how about something larger than infinity?  Next we’ll try the set of all irrational numbers.

Again, in case you’ve forgotten, an irrational number is one that cannot be described as the ratio of two integers.  Things like pi and e (base of the natural logarithm) and the square root of two are decimal numbers that never end and never repeat.  So it would take an infinite number of integers (regular infinity) just to represent each one.  And guess what?  In between every rational number, there are an infinite number of irrational numbers.  If you take pi as an example, you could change any one of its infinite digits and come up with an entirely different number.

So guess what?  We just ran out of slots in our egg carton.  Since we can change any one or all of the digits in any particular irrational number (even an infinite number of them) to get a new number; and since there are an infinite number of irrational numbers, even between each rational number; we have waaay too few slots in our infinite egg carton.

The size of the set of all irrational numbers is the first transinfinite number, aleph-one (aleph-zero is another term for regular infinity).  We can even get bigger than this, defining in similar ways aleph-two, aleph-three, etc, even on to aleph-infinity.

So there you go: a quick and dirty (and honestly, not terribly rigorous) introduction on numbers that are actually larger than infinity.  To summarize (again, not very rigorously): infinity plus infinity equals infinity (all positive and all negative integers); infinity times infinity also equals infinity (all rational numbers); but infinity to the power of infinity is bigger (all irrational numbers).