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>Imagine that Achilles notes the position ... and so on

If in logic you assume/imagine something, and reach a contradiction, then your reaction should be to reject that assumption.

In this case we reach the conclusion that Achilles cannot note all the positions.

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That's not adding anything new. The very idea of the thought experiment is that we reach a contradiction and that we therefore should reject the assumption.

The problem, and the whole point of the thought experiment, is that the assumption is tied to Achilles being able to make the distance, so we should ditch that too.

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