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> The number of physical positions you can be in in space is uncountably infinite

You don't know that. Spacetime may be quantized.

Actually, doesn't the notion of Planck length / Planck time already suggest that shorter distances have no physical meaning?



> Actually, doesn't the notion of Planck length / Planck time

Nope, we have absolutely no evidence of that. Space-time may be either discrete or continuous. Based on our current understanding we have no evidence either way.




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