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Zeno's Dichotomy Paradox
(sh.itjust.works)
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this post was submitted on 05 Apr 2024
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Not that anyone cares but I just realized that this is not actually paradoxical and I can prove it mathematically! (I think) Bear with me since I’ve like just barely learned this stuff this week.
Proof Let S be the set of all steps needed to be taken. It can be written as S = {(distance to be traveled)(2^-n^): n in the Natural numbers}. Thus, S shares cardinality with the natural numbers and is countably infinite.
However, time is continuous. Thus, it has the cardinality of the continuum (real numbers) which means any time interval contains an uncountably infinite amount of moments. Let us denote an arbitrary time interval as T.
Because | T | > | S | there is no injection from T to S. Thus if each step has only 1 time value, there will be moments of time left over, and since the hand is not in two places at once we know each step must have its own time value, so this must be the case.
Therefore, when moving in steps like this, one will run out of infinite steps before they run out of moments in time to complete those steps. Hence, any finite distance can be traversed in this way over some bounded interval of time. QED.
Basically, you can traverse any distance in any time interval as long as physics allows you to move at a fast enough speed. Even if it doesn’t, there may be a limit to how fast you can traverse the distance, but it is still bounded. You can traverse any finite distance like this before existence runs out of time.
(I’m still learning. So if there’s an error in my proof please be gentle lol)
Your proof assumes a non-quantum universe. As soon as time become quantized, you actually do have a minimum finite time step, at which point any observed object by necessity must be at rest. This (well, something to this effect) has been proved by experimental data, however it's currently unclear if this quantization is related to our methods of observation or an actual characteristic of time.
Yes, but if the universe is quantum, then there also exists a minimum finite space step. So the fractions never get infinitely small. So you either stop moving in which case of course you never reach the destination; you stopped before you did. OR you take an extra step and surpass your distance by a negligible amount which means you did move all the way.
So even in a quantized universe, the paradox is still false right?
Well, no — if anything you've proven it. The paradox was originally that all objects must be at rest regardless of observed movement, infinitesimality was just the quickest way for ancient Greeks to conceptualize it.
Experimentally, we've observed† that it is possible to "freeze" time for a quantum particle if you measure it before the wave function has the time to fully transition from "origin" to "target" when no intermediary states exist between††, i.e. a quantum object "in movement" stays at rest until one "time-step" passes, at which point it imediately exists at the next point towards the target, where it remains at rest until the next time-step. If you measure the object "between" time-steps, its position will be at the origin point, but because we've now collapsed the wave function, that position is manifested as reality and no other possibilities exist. As a result, a new time-step must pass before it can move — yet, if we measure again, the same observation will be repeated, so the wave function never gets to the target, even though we have declared that the wave function (and therefore the particle) is moving from origin to target.
That's the kind of fuckery we signed up for when physicists discovered the wave-particle duality
† (potentially, there are competing hypothesis)
†† By necessity, a quantized number line follows this condition