In Case of Interval (or More General) Uncertainty, No Algorithm Can Choose the Simplest Representative
Résumé
When we only know the interval of possible values of a certain quantity (or a more general set of possible values), it is desirable to characterize this interval by supplying the user with the ''simplest'' element from this interval, and by characterizing how different from this value we can get. For example, if, for some unknown physical quantity $x$, measurements result in the interval $[1.95,2.1]$ of possible values, then, most probably, the physicist will publish this result as $y\approx 2$. Similarly, a natural representation of the measurement result $x\in [3.141592,3.141593]$ is $x\approx \pi$. In this paper, we show that the problem of choosing the simplest element from a given interval (or from a given set) is, in general, not algorithmically solvable.