Different Sizes of Infinity Let N be the naturals numbers: 0,1,2,3... and let R be the real numbers which are all the natural numbers and decimal numbers. You can think of the real numbers as all the points on a line. Notice that both of them contain an infinite amount of numbers. For example, you can count 0,1,2,3... without ever coming to and end. Similarly you can say 0.0, 0.1, 0.12, 0.123, ... for eternity. Yet, it seems as though there might be more real numbers than natural numbers. A little investigation reveals that there are infinitely many decimal numbers just between 0 and 1. Then you might ask: Are these sets of numbers the same size? But what would that mean since there are infinitely many of both? Suppose you have a group of kids and a bunch of pencils. How do you know if you have enough pencils for all the kids? You could match up each kid with a pencil. After handing out a pencil to each kid, there might be some empty-handed kids with frowns on their faces. From this you can conclude that there must be more kids than pencils. Likewise if there are pencils left over then there must be more pencils than kids. This matching process of giving one pencil to one kid is how we can determine if something is bigger than the other. Similarly, suppose that there are just as many real numbers as natural numbers. Then we can match them together without any leftovers. We can make a list and match each natural number to a decimal number like this: 0
a0 .b0 c0 d0 ...
a1 .b1 c1 d1 ...
a2 .b2 c2 d2 ...
a3 .b3 c3 d3 ... .
where the a,b,c’s are the digits of the decimal numbers. For instance, one of them could be 2.71828183... or 4.000000... Now consider the real number: a0 .b1 c2 d3 ... This is the real number that consists of all the diagonal digits in our above list. But now let us change each of these digits to another number. For example if this number is 2.71828183..., then change the 2 to any other number like 6. Change the 7 to 3, the 1 to 5 and so forth for the rest of the decimal places. Then this example would give us the real number 6.35473642... After we change each digit of the diagonal number a0 .b1 c2 d3 ... we get the number α0 .β1 γ2 δ3 ... This number is different from all the others in the above list. But that means we have found a leftover real number with a frown on its face! This suggests that there must be more real numbers than natural numbers. Even though there are infinitely many of both, somehow the real numbers are bigger in size than the natural numbers. What does this mean? We have two sets with infinitely many things in them, yet one set is bigger than the other. So in other words, there are different sizes of infinity. In fact with a little work, we can take this further and see that there are infinitely many sizes of infinity! 1