Of the two concepts we are setting out to explore, Big Infinity seems to be the harder of the two concepts to get a hold on, probably because of its lack of visible external boundaries. Perhaps we should start by exploring Little Infinity.

I suggested that one way to model Little Infinity might be to look at all the rational numbers between 0 and 1. How many numbers are there between 0 and 1? We can enumerate them in one fashion as fractions, decimals, or parts of numbers. (0.1, 0.2, 0.3, etc.) How many of these numbers would we say there are between 0 and 1? Does it seem that there is an infinite number of fractional parts between 0 and 1?

Imagine a number line that displays this:

This line is finite in length with distinct boundaries, yet within those boundaries it contains a seemingly endless number of possible divisions. As you approach 0 the numbers seem to get infinitely smaller, and as you approach 1 the numbers seem to get infinitely bigger. With integers like 0 and 1 we can count very easily from one number to the next (0,1,2,3,4, etc.). The next question to ask, then is how we do this with our divisions. What number follows 0? What is the next number after 0 or the last number before 1? What implications might this have for Little Infinity?

You might answer that the number directly following 0 might be something like 0.000...01, such that it is a decimal followed by an infinite number of 0's with a 1 at the end. However, I'm not sure that this means very much at all, since we pretty much arrive at the same problem of being unable to count from 0 to 1. If you believe all of this, it looks like you'll have to accept that there is this thing I have dubbed Little Infinity.

Maybe old Zeno wasn't as wrong as you thought.

If this all seems to be the case, then as you approach 1 from the direction of 0 it seems you could continually come closer and closer without being obliged to count to the number 1. It reminds me of the way some parents might count at their children when scolding them, "One, Two, Two...and a half, Two...and three-quarters..." They rarely reach three and are not obliged to count three because they can continually come closer.

Does all of this make you want to believe in infinity more? What implications does this seem to have for numbers or the number line? What does our investigation seem to say about Big Infinity? Do you think there could be larger and smaller infinities, or is the infinite singular in size?