Spenser

The Problem with Infinity

I am posting this here since God is suppose to have an infinite mind and my understanding of infinity is in direct correlation to an argument against his existence; Infinite Regress

I read this in the discussion forum on Theology online and brain froze.

Well guess what? I switched to a BA cause I hate calculus so maybe some one can help resolve this paradox, or at least explain it a little.

It seems it has been stated that we cannot traverse an infinity, it is something that is not possible. Yet, if you take two points and cut them in half, then take half and cut it in half, then take a half of that and cut it in half ad infinum; Does this mean you will eventually reach a smallest unit of space? Or are there really an infinite number of positions an arrow can be in between point A and B.

I guess can we literally dived a unit of space by infinity; and for that matter can we divide a unit of time by infinity?

Help thaw my brain...

Wayne Delia

I hate it when that happens.

Go to google.com and search on "Zeno's Paradox" for a variety of explanations. The bottom line is that it relies on a faulty assumption that the sum of any infinite number sequence is always infinite; turns out it isn't. For example, if the n-th term of an infinite sequence is 1/2 to the n-th power, then the sum of the sequence constructed by n=0 to infinity (1 + 1/2 + 1/4 + 1/8 + 1/16 + ...) converges to 2.

WMD

Llyricist

Not sure but I think that "infinite divisibility" is actually limited by Planck's constant. That there is a distance or "size" that can be no smaller.

Pyrrho

Wayne Delia gives a good suggestion, but here is the idea in brief:

Imagine a length of one inch. Now, if you divide it in half, you have:

1/2 + 1/2

for the total length. Now, take the second half, and divide it in half, for:

1/2 + 1/4 + 1/4

(because you didn't divide the first half in half, only the second half; and remember, we are talking about a total of 1 inch).

Now, divide the second half of the second half in half (that is, the second "1/4" above should be divided in half), to get:

1/2 + 1/4 + 1/8 + 1/8

Now, continue the above pattern infinitely, to get:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 ...

Notice, we have an infinite sequence of numbers that we are adding. Now, remember what we started with? The total length is 1 inch. The sum of an infinite sequence of numbers can be finite, as the above illustrates.

The same can be done with time. Imagine 1 second. Then divide the second into two parts & etc.

For a more full explanation, if one is desired, lose your dislike for Calculus and take a couple of classes.

stretch

Try picking up a copy of Paradoxes from A to Z by Michael Clark. There's lots in there that you'd probably find useful.

Jobar

Hmmm... let's move this to S&S.

Lobstrosity

Currently this isn't true. Space is not quantized. String theory could prove to impose limits on the fundamental scale with which we could view the universe, however.

With regards to the opening post of this thread, the main thing you're not taking into account is that the amount of time it takes to traverse a given interval goes to zero as the interval size goes to zero. Specifically, in order to have an infinite quantity of intervals in the overall distance, the interval size must go to zero, meaning that the time you take to traverse each interval is going to zero. This is why the sum of times can remain finite (just like the sum of an infinite number of infinitely small intervals remains the finite overall distance you are breaking up).

An interesting problem is one I saw posted here a few months back. Drop a ball from a certain height h (say 1m). Every time it hits the ground it bounces but loses half its energy in the process. Pretending this is an ideal, classical universe, how many times does it bounce? How long does it bounce before it comes completely to rest?

The answer is a surprising one. First and foremost the ball will bounce an infinite number of times. If you plot energy as a function of bounce you can see that no matter what bounce its on the ball will always have some nonzero energy left. The interesting part, though, is that you find the ball does in fact bounce all infinity times in a finite timespan. After 2.61 seconds it is completely at rest. It seems impossible that something could bounce an infinite number of times in a finite timespan, but I assure you it is not (at least in an ideal classical universe).

Tenpudo

Dammit, Lobstrosity, you made me recall all those structures we dealt with in my first semester Calc. class that have finite volume but infinite surface area...

Wayne Delia

Would it help if we reduced the number of dimensions? How about a 2-dimensional polygon with finite area and infinite perimeter? The pathological "snowflake curve" is constructed by starting with an equilateral triangle, and constructing a smaller equilateral triangle on each side using the middle third of each side as a base, removing the base in the process (now it looks like a Star of David). Put the resulting polygon through the same process - construct a smaller equilateral triangle on each side using the middle third of each side as the base, removing the base. Iterate that process infinitely many times. The resulting polygon looks like a snowflake, with infinite perimeter and a surface area of two-fifths times the square root of three times whatever the length of a side of the original equilateral triangle was.

No kidding. You can draw a line infinitely long on a postage stamp.

WMD

Lobstrosity

Right, that is one of the coolest things I have ever seen. Take the curve defined by y = 1/x on the interval from x = 1 to infinity and rotate it around the x axis. The surface area of the shape defined by this action is infinite while the volume enclosed by that surface area is finite (in fact, it is exactly pi/2, I believe). As such, you could completely fill it with paint but you'd never have enough to coat the surface.