Positions in Infinite Space

mathman

We seem to have reached a point where we are trying to define terms (like traveling forever). The math itself is clear.

netzweltler

A math example:

The union of the segments [0, 0.5], [0.5, 0.75], [0.75, 0.875], ... is [0, 1) and not [0, 1]. If I am drawing these segments of the unit interval I am drawing infinitely many segments, but I am not drawing the last point 1.0 (which in some sense could be named the [itex]\omega[/itex]th segment).
If the drawing of one segment takes a second, I am drawing these segments forever, and I am not reaching a point infinitely apart (in some sense point 1.0).

That's what I meant, when I was asking:

mathman

The fallacy in your analogy is that each step takes a fixed amount of time, no matter how small the interval.

I don't see the connection to the original question where (as I have already said) you need to give a precise definition of what you mean.

netzweltler

To see the analogy it might be helpful to biject the previous example to the trip of the set of mathematicians who haven't reached their destination. Let's say

1. travelling the first meter → drawing the segment [0, 0.5]
2. travelling the second meter → drawing the segment [0.5, 0.75]
3. travelling the third meter → drawing the segment [0.75, 0.875]
...

Do we have two different definitions of infinite travelling?
1. Travelling between two points infinitely apart
2. Passing every finite meter/segment of infinitely many meters/segments

In the first case we arrive at the point infinitely apart. In the second case we don't. According to the first definition the set of mathematicians who haven't reached their destination is empty at arrival at the point infinitely apart. According to the second definition the set of mathematicians who haven't reached their destination is non-empty for the whole infinite trip, and so it might be valid to state, that the mathematicians of this set are on an infinite trip, thus questioning that there are only finite distances in infinite space.

Number Nine

This whole conversation seems to have gone somewhere where I don't know where it is. First, the original question: The travel time of every mathematician is finite. We've settled that. It's done. As for this comment...

I'm not sure what you're doing here, but it seems to be some sort of reference to Zeno's paradox, which is, of course, just a problem of infinite series. The travel times corresponding to each of those distances (even carrying on the pattern into infinity) sums to a finite value. More specifically...[tex]\sum_{i=1}^\infty \frac{1}{2^n} = 1[/tex]
If that's not what you're talking about...then I have no idea.

We've been discussing "infinite travelling" in terms of travel times, and the issue has been settled. As for your two points...

1) There are no such points. The euclidean metric assigns to every two points in the plain a finite distance.
2) I really have no idea what this is supposed to mean.

netzweltler

To get an idea of what I mean it might help to clarify this first:
I am shifting the line [0, 1] in infinitely many steps to the left

step 1: I am shifting the line [0, 1] to position [-0.5, 0.5]
step 2: I am shifting the line at [-0.5, 0.5] to position [-0.75, 0.25]
step 3: I am shifting the line at [-0.75, 0.25] to position [-0.875, 0.125]
...

At which position is the line after infinitely many steps?

Number Nine

[-1, 0]

netzweltler

At which step does the leftmost point of the line move to position -1?

SteveL27

I don't follow that. It's like taking the sequence 1/2, 1/4, 1/8, ... and asking what's the value of the sequence "after infinitely many steps." If you're using the phrase to mean "what is the limit of the sequence as n goes to infinity," then the answer is zero. But there is no point at which the value of the sequence is actually zero; and that's a huge conceptual subtlety in a discussion like this. Isn't it?

Number Nine

it's not a sequence, it's a series. First he's moving 0.5 to the left, then 0.25...

That's not how infinite series work. The limit as the number of steps go to infinity is [-1, 0], which is what you asked. If you meant something different, then elaborate.

netzweltler

The number of steps is not going to infinity, the number of steps is infinite. Think of it as an actually infinite list of steps. None of the steps on this list is shifting the line to postion [-1, 0]. I can understand, that there are different notions of infinite travelling. That's what I meant before. According to your notion the line is actually arriving at [-1, 0]. According to the list above the position of the line is not defined after infinitely many steps, but the line has done an infinite trip.

Number Nine

No, the line is [-1, 0] "after infinitely many steps". This a very elementary infinite series.

Honestly, I have no idea what it is that we're supposed to be talking about anymore. All of the initial questions about "infinite travelling" have been answered, but you seem to have spontaneously created some sort of brand new definition that you're not sharing with anyone. Please clearly explaining what it is that you're asking.

netzweltler

step 1: at t = 0 I am shifting the line [0, 1] to position [-0.5, 0.5]
step 2: at t = 0.5 I am shifting the line at [-0.5, 0.5] to position [-0.75, 0.25]
step 3: at t = 0.75 I am shifting the line at [-0.75, 0.25] to position [-0.875, 0.125]
...

All shifting is done before t = 1. No action is done at t = 1. Countable infinity doesn't allow a last shift at t = 1, which clearly is needed for point -1 to be covered.

If this is my private math solely, further clarification doesn't make sense.

Number Nine

...what? Where does "countable infinity" enter into all of this? And what does it have to do with "moving at t=1"? You aren't even bothering to explain what it is that you're doing or asking, and it's making it very difficult to answer any of your questions. What does this have to do with "positions in infinite space"?