Is Distance Between Positions in Infinite Space Always Finite?

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The discussion centers on whether the distance between any two positions in infinite space is always finite. It highlights that in a finite-dimensional space, such as Euclidean 3-space, distances between identifiable points are finite. However, in an infinitely large universe, certain points may be unreachable due to the universe's expansion outpacing the speed of light, leading to the conclusion that some distances could effectively be infinite. The conversation also explores the concept of infinite mathematicians traveling to different points, emphasizing that while their travel times are finite, the notion of "traveling forever" can imply different interpretations regarding reaching infinitely distant points. Ultimately, the distinction between finite distances and the concept of infinite travel is crucial in understanding the nature of space.
  • #31
netzweltler said:
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"?
 

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