Virgil Șerbănuță edited untitled.tex  over 8 years ago

Commit id: 34a1f6fb1ebd6233ee318ae8a8f0ea8d1d1d7344

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Now, let us return to the issue of observable models being finite or infinite. With an finite alphabet (and even with an infinite but countable one), only a countable set of models have a finite observable description. Then the "has a finite description" proposition is a zero-probability one, so either our universe is created, or at any point in time there will be an infinite number of things that we didn't manage to model about our universe but we think that they are important. [TODO: It's probably better to say that there will be an important part of our universe that we can observe but can't model. Also, since we chosed the precision and prediction error in an arbitrary way, this part that can't be modelled is visible at any "zoom" level.]  Let us now do the following:  \begin{ennumerate} \begin{itemize}  \item fix $\epsilon \gt 0$ and say that we care about measuring things with a precision $\epsilon$;  \item restrict ourselves to universes in which we can measure things with real numbers;  \item ignore things which happen rarely, even if we can measure that they happened using the $\epsilon$ precision. [TODO: this is included in the fraction $f$ below]  \end{ennumerate} \end{itemize}  This is a bit hand-wavy, but we could use any reasonable definition of "measuring" and "happen rarely". Then we could say that the important things are the ones which we can measure with a precision greater than $\epsilon$ and which do not happen rarely. Let us also fix an arbitrary time length $t>0$, an acceptable error $\delta \ge 0$ for our predictions and let us denote by $f$ with $0 < f \le 1$ a fraction of the world where we can make predictions using the given time $t$ and the acceptable error $\delta$.  Then, if the world is not created, then using any continuous distribution the probability of having a finite description with which we can do this is $0$, which means (with probability 1) that $f=0$. This happens even if we don't care about predicting things precisely (i.e. making $\delta$ larger) or if we restricts ourselves to small time scales (making $t$ smaller). We can only make prediction for a tiny part of the universe, so tiny that it's practically nothing ($f=0$).