The Spaceship

A spaceship travels in a straight line. It doubles its speed after half a minute, doubles it again after another quarter of a minute, and continues successively to double it after half of the last interval. Where is it at one minute? It is neither infinitely far away, nor at any finite distance either.

It is not infinitely far away, since there is no such place. But it cannot be at any finite distance from its start after one minute, since that would involve an inexplicable spatio-temporal discontinuity in its existence. For in that minute it could not trace a continuous straight-line path through space and time to any point a finite distance from its start if it were to satisfy the description above: any
finite distance from the start, however far, is reached before the minute is up.

The only way to avoid this incoherence would be for it to shrink gradually into nothing, say by halving every time it doubled its speed. In any case, if it did not shrink like this, it would have to be travelling infinitely fast at one minute. So the only admissible answer is: nowhere. (If, on the other hand, you allow it to be spatiotemporally discontinuous, it could be anywhere.) The paradox is an invention of Benardete’s.

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