There are infinitely many whole numbers, clearly: If you have a finite list of them, take the biggest one and add one to get a new one. There are also infinitely many real numbers in bigger than or equal to 0, but less than or equal to 1: any positive decimal number less than one is in there, for instance, along with an infinity of numbers that can't be written as finite or repeating decimals (e, pi, the square root of two, etc.).

There are more real numbers than there are whole numbers, in the sense that the real numbers cannot be put into one-to-one correspondence with the whole numbers. Any way in which you assign one real number to one whole number cannot help but leave a real number without an assignment. This is a thing that pisses off fundamentalist Christian textbook authors. You should read that article, because it's nuts. Apparently, certain fundamentalists reject modern math; that is, they reject any system of mathematics which allows for different "sizes" of infinity, since that is an affront to God, who is the one and only infinity. I've written about this sort of thing before; right now I want to prove to you that there are more real numbers than there are whole numbers, in a very clear sense. There is more than one infinity, for sure, and it's not even very difficult to prove, and it's downright embarrassing to insist otherwise, and those dudes are total morons.

We will demonstrate that there cannot even be a one-to-one association of whole numbers to decimal numbers less than one that include only the digits 0 and 1 (so-called binary decimals), and we will do that by showing that no matter how you choose to associate whole numbers to binary decimals, there is always a binary decimal without a whole number partner. In this sense, there are more binary decimals than there are whole numbers, and thus it is possible for two infinite sets to be of different sizes. There is more than one infinity.

Suppose we create such an association: For each whole number, there is a corresponding binary decimal number. Construct a new binary decimal as follows: The nth digit of it is the opposite of the nth digit of the binary decimal associated to the whole number n.

For example: Consider the number 5. We're assuming that the number 5 has been associated to a certain binary decimal. The 5th digit of this binary decimal is either 0 or 1; if it's 0, the 5th digit of this new number we're constructing will be 1, and vice-versa.


Therefore, the binary decimal we've constructed is not equal to the binary decimal associated to 5; it differs in its 5th digit, at least. Similarly, it can't be equal to the binary decimal associated to any whole number; it will differ from the number associated to n in its nth digit. This new binary decimal has no whole number associated to it! Quite literally, the binary decimal numbers are uncountable; any attempt to do so will miss one, as we've demonstrated. The infinity of whole numbers is not enough to account for all the binary decimals.

QED: There are infinities of different sizes, and fundamentalist Christianity is for idiots.