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

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

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

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

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

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

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

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

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

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?