Wednesday, January 28, 2009

Little Infinity

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?


Pocket Size said...

"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?"

It's interesting that you ask about the number line itself. Being a visual aid, the number line can be drawn at any scale we want. Technically, it could be drawn even at a varying scale, with different distances between different integers. Maybe one inch between 1 and 2, but three inches between 2 and 3, and then two inces between 3 and 4, for example. It would look funny, but since every section represents an infinite number of divisions, who's to say which lengths are right and wrong? Hmm, this could make abstract art way more fun...

Anyway, off the top of my head it doesn't seem to say much about Big Infinity, because it seems to me that you could take your 0 to 1 number line and turn it inside out and you'd have Big Infinity. Kinda like the difference between a veggie calzone and a veggie pizza (oh here I go making myself hungry). You have the same ingredients and the same flavor in both, but the difference is how it's arranged. In the case of infinity, the difference is in where you put the decimals and 0's.

That may or may not have made any sense, but there it is.

B said...

It's true that they seem very much alike. I'm wondering about the numerability of various infinities. The different spaces between numbers on a number line are a matter of scaling, but the difference between any two numbers is always the same (it is 1).

Do you suppose Big and Little Infinity are such different things? Or are they the same sort of thing? It's interesting that you say the are just in a different order. Do you think that big infinity does have a boundary somewhere, like the bounded edges of Little Infinity? Or do you suppose that the seeming boundaries of Little Infinity are not boundaries at all? If this were the case, it might be that Big Infinity is just a kind of scaled Little Infinity regarding which we are not aware of the boundary (or at least the top boundary, since we could set 0 as a lower bound).

Pocket Size said...

I think that in Little Infinity, the boundaries are false. Since the space in between is infinite, the boundaries mean nothing.

B said...

I'm intrigued by your idea. I'm not so sure that the boundaries are meaningless, but maybe they mean less than I originally thought. In the example of 0 and 1 we have a seeming infinity of rational numbers between them. Somehow though, this infinite set of rational numbers is constrained in their value by the boundaries.

The numbers cannot be less than the lower bound of 0, (although perhaps that is a bad example since it brings up some of the oddity of 0) and cannot be greater than the upper bound of 1.

It seems to me that the infinite set of numbers isn't entirely without relation to the boundaries.

This also raises the question of whether the boundaries of 0 and 1 are part of the rational numbers in question. Your suggestion seems true to me if we disclude the boundaries from the infinity we are discussing.

Does your suggestion that the boundaries are false seem to you to indicate that there is no difference between Big and Little Infinity? Or is there still a difference between them?

Kat said...

I don’t know this might beneath what your thinking about or perhaps something that has nothing to do with you analyzing infinity.

However technically, in my perspective I guess wouldn’t the measurement for infinite be time. So infinite doesn’t stop until that person/thing/whatever is dead/stopped/frozen.

An in that case no perspective on infinite would be meaningless. I’ll take the example of the pizza/calzone. Yes you could arrange the fixings in the most “meaningless ways” however their really not that meaningless. If you put the cheese under the sauce its going to taste different. Or maybe you put all the veggies in the middle and some pepperoni on the out side, that will taste different then distributing the items evenly. And you could even go to the extreme and just roll the whole thing up into a ball and throw it in the oven and see what comes out.

Again I guess a tribute to the fact that the human mind can’t take in the full extent of infinite. So therefore the illusion of infinity is only the perspective of what you as an individual can handle…. Mentally. (example: infinity to you is a little 0 to 1 line. Mine is equal to time)

Oh and P.S. technically wouldn’t you have a 3D space of a like between zero to one on your example?

P.P.S…… I probably have no idea what im talking about. But it thought the topic was interesting…. And I like little graphs