Positions in Infinite Space

[netzweltler]

Is the distance between any position A and any position B in infinite space always finite?

 

[mathman]

You need to define position precisely. What do you mean by infinite space - finite dimension with unlimited size or infinite dimension?

For example in 3-space, if A = (a1,a2,a3) and B=(b1,b2,b3) where a's and b's are finite numbers and the distance is Euclidean, then the distance is finite.

 

[netzweltler]

How does it apply to an infinitely big universe? If I am starting from position A is there possibly a position B in this universe which I can never arrive in finite time?

Or is it possible to arrive at any point of this space in finite time, no matter which point I am starting from?

 

[theorem4.5.9]

There are mathematical spaces which have components that are "infinite" distances apart. An easy example is to take two compact (closed and bounded) disconnected components in $\mathbb{R}^2$. Define the distance between any two points to be the minimum length of all paths that live in the two regions, and connect the two points together.

Because the two regions are disconnected, there are no paths connected points between the two. Hence any points in different regions are infinite in distance.

Your second post is about something more physical, as there is no notion of "time" in the example I posed. I think most people assume the universe is simply connected, which implies every two points to be a finite distance apart. However, the rate of expansion of the universe increases in proportion to distance. That means that if a point is far enough from you, the distance between you and that point will be increasing faster than an object can travel at the speed of light. In effect, there are points in the universe that cannot be traveled to, and in effect, the observable universe is getting smaller.

Disclaimer: I may be completely wrong on the physics, but it's my current understanding.

 

[netzweltler]

To clarify this: I do not mean an expanding universe. I mean an actual infinite universe (let's assume the space our physical universe is expanding in is infinite). I just don't know what the correct mathematical model is.

 

[mathman]

To clarify this: I do not mean an expanding universe. I mean an actual infinite universe (let's assume the space our physical universe is expanding in is infinite). I just don't know what the correct mathematical model is.

The simplest model is ordinary 3-space, using Euclidean geometry. Then every point with identifiable coordinates is at a finite distance from any other point with identifiable coordinates.

 

[netzweltler]

The simplest model is ordinary 3-space, using Euclidean geometry. Then every point with identifiable coordinates is at a finite distance from any other point with identifiable coordinates.

What am I doing wrong in this thought experiment:

Infinitely many mathematicians are starting simultaneously from position A in their space ships. Each mathematician has chosen a different destination (a different position B). So there is one mathematician for each possible position B in space. Whenever a mathematician arrives at his destination there are infinitely many other mathematicians who haven't reached their destination yet. This is _always_ true. So, there are _always_ mathematicians who haven't reached their position B.
Does it mean, that _never_ all of them do arrive at their destination?

 

[mathman]

What am I doing wrong in this thought experiment:

Infinitely many mathematicians are starting simultaneously from position A in their space ships. Each mathematician has chosen a different destination (a different position B). So there is one mathematician for each possible position B in space. Whenever a mathematician arrives at his destination there are infinitely many other mathematicians who haven't reached their destination yet. This is _always_ true. So, there are _always_ mathematicians who haven't reached their position B.
Does it mean, that _never_ all of them do arrive at their destination?

Between any two points in space, the distance is finite. However there is no upper limit to distances between pairs of points, which is the essence of your example. I'll assume they are all traveling at the same speed.

 

[netzweltler]

Between any two points in space, the distance is finite. However there is no upper limit to distances between pairs of points, which is the essence of your example. I'll assume they are all traveling at the same speed.

For simplicity yes, they are traveling at the same speed. Do you agree, that there are mathematicians on a non-ending trip in this setup?

 

[Number Nine]

You may have an easier time understanding this if you understood the notion of a metric, which is how distances are defined in a mathematical sense. Take the plane as an example: A metric on the plane is a function that assigns, to every pair of points, a real number (subject to certain restrictions), which is the distance between the two. Pick any two points in the plain: their distance is then a real number ("infinity" is not a real number). You can then calculate the travel time based on that number; it my be arbitrarily large, but it is necessarily finite.

 

[netzweltler]

You may have an easier time understanding this if you understood the notion of a metric, which is how distances are defined in a mathematical sense. Take the plane as an example: A metric on the plane is a function that assigns, to every pair of points, a real number (subject to certain restrictions), which is the distance between the two. Pick any two points in the plain: their distance is then a real number ("infinity" is not a real number). You can then calculate the travel time based on that number; it my be arbitrarily large, but it is necessarily finite.

I can reach every point with identifiable coordinates - out of (-oo,+oo), no matter how long it takes. My question is if infinite space (or plain) is solely made out of these identifiable points. By definition it is. I am just wondering why the example above appears to question that.
Every mathematician has chosen one of these identifiable coordinates as his destination. So, every mathematician will arrive at his destination. But there is always a set of mathematicians left still travelling. As long as you are one of the mathematicians of this set,
you are on an infinite trip. And the set never disappears.

 

[Number Nine]

But there is always a set of mathematicians left still travelling. As long as you are one of the mathematicians of this set,
you are on an infinite trip. And the set never disappears.

This is sort of misleading. What's happening is that the distances (and thus the travel times) are unbounded, and so can be arbitrarily large. Nevertheless, every mathematician's travel time is finite.

 

[mathman]

Simple example: set of integers. Every integer is finite, but there is no upper limit.

 

[theorem4.5.9]

For simplicity yes, they are traveling at the same speed. Do you agree, that there are mathematicians on a non-ending trip in this setup?

No, there are no mathematicians who travel for all time. Every mathematician has a finite distance to travel, and will arrive in finite time.

It seems your confusion parallels a lot of students confusion between convergence vs uniform convergence (say for example of sequences of functions). The point is that in your example, any mathematician's trip has nothing to do with any other mathematician's trip. When you look over the entire scenario, you have arbitrarily large numbers, but numbers nonetheless (as opposed to actually having a point at infinity as in the extended plane).

 

[netzweltler]

What about my statement, that the set of mathematicians who haven't reached their destination does exist forever? Is this statement true or false?

 

[mathman]

What about my statement, that the set of mathematicians who haven't reached their destination does exist forever? Is this statement true or false?

My comment about the integers applies here. After any finite number there are an infinite number of larger numbers. For your question after any finite time there are an infinite number who are still going.

You need a precise definition of the set you are talking about.

I'll illustrate by using set theory. Let A(n) = {k|k >n}. Each A(n) is an infinite set. However the intersection of all A(n) is empty.

 

[netzweltler]

My comment about the integers applies here. After any finite number there are an infinite number of larger numbers. For your question after any finite time there are an infinite number who are still going.

You need a precise definition of the set you are talking about.

I'll illustrate by using set theory. Let A(n) = {k|k >n}. Each A(n) is an infinite set. However the intersection of all A(n) is empty.

Makes perfect sense to me from a set theoretical point of view. Is it possible, that "travelling forever" simply means "travelling to each position finitely apart" and not necessarily, that a position infinitely apart is reached?

 

[mathman]

Makes perfect sense to me from a set theoretical point of view. Is it possible, that "travelling forever" simply means "travelling to each position finitely apart" and not necessarily, that a position infinitely apart is reached?

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

 

[netzweltler]

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

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 $\omega$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:

Is it possible, that "travelling forever" simply means "travelling to each position finitely apart" and not necessarily, that a position infinitely apart is reached?

 

[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. traveling the first meter → drawing the segment [0, 0.5]
2. traveling the second meter → drawing the segment [0.5, 0.75]
3. traveling 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...

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

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...$$\sum_{i=1}^\infty \frac{1}{2^n} = 1$$
If that's not what you're talking about...then I have no idea.

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

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]

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?

[-1, 0]

 

[netzweltler]

[-1, 0]

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

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]
...

 

[SteveL27]

[-1, 0]

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]

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 difference in a discussion like this. Isn't it?

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

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

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]

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]
...

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]

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.

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]

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.

...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"?