Recent content by netzweltler

If we allow only finitely many copies of a line segment s in [0, 1], where the size of s is an element of (0, 0.5], how can we allow infinitely many line segments { [0, 0.5], [0.5, 0.75], [0.75, 0.875], ... } in [0, 1], where the size of each line segment is an element of (0, 0.5]? We wouldn't...

That's the sense that I mean - the infinite sense - when I am asking for the limit case of this task:

Which mathematical model are you talking about? The task is to design a process that enables us to draw a line from point 0 to point 1 in 1 second in infinitely many steps. The pen is not allowed to change direction. Point 1 is the last point covered by the line. There is no last step in this...

I think I can understand the concept of limits, but I am not that sure about it. If I try to apply the concept of limits to the task I guess the (most) correct assumption is, that all segments in [0, 1) are named #1 in the limit case. But what happens to the other segments which are always...

We don't use "biject"? :bugeye: Oops! Sorry. If we don't stop using #1 all segments in [0, 1) will be named #1 at t = 1, right? What about all the other segments (#2, #3, ...) at t = 1? I mean it like this: t = 0: move pen from 0 to 0.5, naming it segment #1, [0.5, 0.75] is #2, [0.75...

We know we can biject the segments of size 0.5, 0.25, 0.125, … to the segments of size 0.25, 0.125, 0.0625, …, furthermore we can biject these segments to the segments of size 0.125, 0.0625, 0.03125, …, and there is no end to it. So, we can use index #1 for segment 0.5. We can as well use index...

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...

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...

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

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...

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...

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 \omegath...

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?

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

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...