The discussion centers on the concept of distance in infinite space, specifically whether the distance between any two positions A and B is always finite. It is established that in Euclidean 3-space, if A and B have finite coordinates, the distance is finite. However, the conversation explores scenarios in an infinite universe where certain points may be unreachable due to the universe's expansion, leading to infinite distances between disconnected regions. The participants clarify that while distances can be arbitrarily large, each individual journey remains finite.
PREREQUISITESMathematicians, physicists, and students of mathematics interested in the concepts of distance in infinite spaces, topology, and the implications of cosmological theories on spatial relationships.
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]
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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.