This question is related with the Cardinality of the continuum and Zeno's paradox about Achilles and the tortoise regarding the concept of infinity. I would like to know if the following guessings are valid or there are flaws on the train of thoughts. It is a paradox regarding the measurement of a finite distance:

(Caveat: I will try to clarify the question as much as possible, but I understand that it might not fit the standards of MSE. If finally I am not able to do it correctly I will remove the question to keep clean the open question list).

  1. We, as humans, measure distance in specific discrete units: steps, meters, half meters, centimeters, etc. When we do so, we can measure a finite distance in some finite time units. E.g.: we can measure that the distance to the door is $12$ steps and we can verify that calculation in less than (just an example) $10$ seconds.

  2. But this is because we are measuring by using discrete distance units. A meter, a centimeter, etc. But what happens if we use an infinitely accurate unit of measurement? My guessing is that we cannot finish measuring the finite distance, because we need an infinite time to measure exactly the distance when the accuracy of the distance is forced to be infinite. My guessing ($G1$): the accuracy (length) of a distance measurement would be in that case equivalent to the cardinality of the continuum, $c$. In other words: humans cannot afford to be infinitely accurate.

  3. Now imagine a non human being that is able to measure using an infinite distance unit in a finite (for the being, not for us humans) discrete unit of time. This being can measure the specific relative position of a point with infinite accuracy respect to another reference point in a finite time unit (of the being).

  4. My guessing is that ($G2$) if this being is able to measure in some finite units (for the being) of time the finite (again for the being) distance, that time frame is measured in the "cardinality of the continuum" $c$ units of time. So this being can measure in discrete infinite time units a discrete distance measured in an infinitely accurate distance unit.

So my question are:

  1. Are the guessings $G1$ and $G2$ valid or there are flaws in the train of thoughts?

  2. Let us imagine that a human is watching this being to measure the distance, since the very moment the being starts to measure it. My guessing is that the human will not see the being finishing the measuring of the distance, but the being will surely finish to measure sometime in his different discrete units frame of time. So basically the clocks of the human and the being are different and an action that for humans will take an infinite time for the being will be a finite time. Is this correct? Thank you!

  • $\begingroup$ Sensors, as built by humans, can indeed measure magnitude with arbitrary small resolution accuracy, through physical principles, which are not limited by preestablished units, but through the very physical limits of the effects they are based on (wavelengths, cristalline matter dimensions, molecular dimensions, physical effects distances, etc.). Much before that, the physical perturbations limit your measurement in the form of noise. Hence i think the question is too... theoric... $\endgroup$ – Brethlosze Nov 17 '17 at 2:15
  • $\begingroup$ @hyprfrcb thanks for showing your thoughts! IMHO maths are involved with the physical reality and also with the world of theoretical ideas that are not reflected necessarily in the physical reality. Sensors, even if they can measure with arbitrary resolution, do not have infinite resolution, but we can idealize and verbalize concepts regarding the infinity. And usually paradoxes are more or less out from the reality (as it happens for instance with Achilles and the tortoise or the Banach–Tarski paradox). So I think there is space for them too. $\endgroup$ – iadvd Nov 17 '17 at 2:23
  • $\begingroup$ Compare a human making a measurement, and a sensor making the same measurement. The human take some time (some finite time) at his own resolution in giving a human measure (with human units resolution). And at the same time, a sensor measure take a physical effect (in zero time) to occur for giving a physical measure (in infinite resolution). So we can argue a sensor as a closest example of that infinite being, and we humans see them measure objects at zero time with infinite resolution. So we indeed see the sensor starting and finishing a measure. $\endgroup$ – Brethlosze Nov 17 '17 at 2:27
  • $\begingroup$ So G1 is valid because human measure a value through a converging truncated series, G2 is valid because infinite being only need to observe the right value, Q1 is OK, and Q2 is wrong. $\endgroup$ – Brethlosze Nov 17 '17 at 2:32
  • $\begingroup$ @hyprfrcb what I do not see clear is that the sensor is measuring in infinite resolution. I think the hardware of the sensor is not perfect, it is discretizing the physical measure due to the accuracy limits of the hardware of the sensor. $\endgroup$ – iadvd Nov 17 '17 at 2:34

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