Infinite Numbers

When we count things in our everyday life we normally use the natural numbers   0,1,2,.....   This is all we need to say things such as  "I have three pencils",  "the Federal budget is over three trillion dollars"  and so on.  So, is this enough to count everything?
No, because it is not enough to, for example, count all the natural numbers!  Consider the following Generalizing, we get that for any natural number n there are n numbers less than, but not equal to, n,  namely all the natural numbers in the set {0,1,2,...,n-1}.  But then there can't be a natural number  that is the number of natural numbers for there are already p numbers in the set {0,1,...,p-1}  and so there must be at least  p+1 numbers in the set {0,1,2...,p-1,p} which is, at best, just a subset of the set of all natural numbers.

Thus, if we want to talk about the number of numbers in the set of all natural numbers we need to introduce a new number which is not a natural number.   Lets call this number  ( the Greek letter omega, pronounced Oh-may-gah with the accent on the second sylable).  Now we can say, "there are  natural numbers".  This new number. , is not a natural number but what is called an infinite number.

Now saying that there are  natural numbers doesn't tell us all that much since we defined  to be precisely the number of natural numbers.   But it could mean something to say that there are  elements in some other set such as the set of all rational numbers or the set of all real numbers.   As it turns out there are exactly 
rational numbers but there more than  real numbers.  It follows that the number of real numbers is a bigger infinite number than  and thus that there is more than one infinite number!