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The task the OP presents is similar to the Achilles and the Tortoise presented by Zeno in Plato's Parmenides. A task which can be completed is divided into an infinite series of smaller tasks which now cannot be completed.
Nick Huggett describes this as a "supertask" and offers the following resolution based on M. Black's "infinity machines" (‘Achilles ...
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In my opinion, the underlying fallacy of your argument is that you assume that because you have described the motions as divided into an infinite sequence of sub-motions, then somehow they actually ARE divided into an infinite sequence of sub-motions. This is not true; there is only a pair of objects moving continuously. Just because you can mark an ...
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Philosophically speaking, a paradox (like the version of Zeno's paradox outlined here) is not meant to show that there is something wrong with the world. It is meant to show that there is something wrong with our understanding of the world. Zeno was not disputing that motion was possible, since it self-evidently is. Zeno was pointing out that the mathematics ...
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If the question at hand is one of modeling, you will likely keep them all as a "toolbox" and use whichever one lends itself best to solving a particular problem. A good example is Newtonian, Lagrangian, and Hamiltonian mechanics. They are all "correct" but they are all also only models of what is "actually happening." They all give the same answers and are ...
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Taking your example of the two "equivalent" theories (which we will assume are both mathematical models), one will be favored over the other for the following reasons:
1) One model makes testable predictions that the other one does not,
2) One model successfully accounts for observations and data collected in the past, which the other one does not, and/or
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