Some Properties of Infinity

How big is infinity? There is a surprising answer to this question: there are actually multiple sizes of infinity. Actually, there are infinity sizes of infinity. Let us look at a few examples to illustrate these points. First of all, how many numbers are in the intervals [0,1], and [0,2]? Obviously, there is an infinite number of numbers, but in some sense, the two intervals contain the exact same number of numbers. This may seem counter-intuitive, because the length of the second segment is two times longer than the length of the first segment. However, one simple bit of reasoning will show the truth of the statement. Given any number in the first interval, if we double that number, it will be a unique number in the second interval. Similarly, given any number in the second interval, if we halve it, it will be a unique number in the first interval. What we have just done is establish a one-to-one correspondence between the two intervals. Now if the two intervals did not have an equal number of numbers, then we could not establish a one-to-one correspondence between the two, because there will always be at least one number in one of the intervals that is left over and does not correspond to any number in the other interval. This example illustrates the fact that the size of infinity does not depend on any finite quantity, such as length. Another similar example is that if we take every positive even integer and halved it, it would be the integers, and if we took every positive integer and doubled it, it would be the even integers. This shows that the sets {1,2,3...} and {2,4,6...} have the same number of numbers.

Now onto different sizes of infinity. It seems pretty obvious that there are more real numbers (numbers that can be expressed as infinite decimals) than integers, but the result takes a little bit of subtle reasoning to prove. Say there were as many integers as reals. We could then list out all the real numbers starting from the first real number, to the second real number, to the third, and so on. All the real numbers can be expressed as infinite decimals. We could then choose a digit different from the first decimal place of the first real number, a digit different from the second decimal place of the second real number, and so on. We then use these digits to construct a real number with the first decimal place as the first number chosen, second decimal place as the second number chosen, and so on. This real number would then differ in at least one spot from all of the listed real numbers, which means that no matter how we try to list in order out all the real numbers, there will always be a real number not listed. This implies that the number of real numbers is greater than the number of integers.

One last, and more counter-intuitive example is the fact that there are just as many integers as there are  rational numbers (numbers that can be expressed as p/q, where p and q are integers). The proof of such as result is very hard to explain without a picture:

Every single rational number is touched by the line; the line essentially amounts to an ordered list of rational numbers. Therefore the number of integers is actually just as great as the number of rationals. This, and the preceding example, establish the concept of different sizes of infinity.