Can Space Be Infinitely Divided?

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How many times can I half the distance between my hands? Assume perfect coordination and the ability to localize my palms to the quantum level. 15 halvings get them to within a cell’s width. 33 to within a single atom, 50, and they’re a proton’s width apart. Half the distance 115 times and they’re a single Planck-length apart – 1.6×10^-35 meters. Surely we can keep going – .8, .4, .2 x10^-35 m? Bizarrely, those distances might not even exist in any meaningful way.

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