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I don't think time and measurement precision are necessarily related in that way. You can measure weight with increased precision by using a more precise scale, without increasing the time it takes to do the measurement.

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The real point is that it takes infinite energy to get infinite precision.

Let me add that we have no clue how to do a measurement that doesn't involve a photon somewhere, which means that it's pure science fiction to think of infinite precision for anything small enough to be disturbed by a low-energy photon.

I'm not making the case that it is possible to make measurements with infinite precision. I'm making the case that the argument "It is not possible to make measurements with infinite precision, therefore we cannot tell if we live in a rational or a real world." is begging the question. The conclusion follows logically from the premise. Unless the argument is just "we can't currently distinguish between a rational and a real world", but that seems trivial.

There are limits to precision there too. The amount of available matter to build something out of and the size you can build down to before quantum effects interfere.

The example was only to illustrate that measurement precision is independent of the time it takes to perform the measurement.

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