## 8 Justification of [PERSON]'s Razor  
We now prove [PERSON]'s razor under the assumptions stated below and compare it to the No Free Lunch myth. The result itself is not novel [9]. The intention and contribution is to provide an elementary but still sufficiently formal argument, which in particular is free of more sophisticated concepts like [PERSON]'s a-priori distribution. [PERSON]'s razor principle demands to \"take the simplest theory consistent with the observations\". _[PERSON]'s razor could be regarded as correct if among all considered theories, the one selected by [PERSON]'s razor is the one that most likely leads to correct predictions._  
Assumptions.Assume we live in the universal multiverse \(\ddot{u}\) that consists of all computable universes, _i.e._, UToE is a correct/true/perfect ToE. Since every computable universe is contained in UToE, it is at least under the computability assumption impossible to disprove this assumptions. The second assumption we make is that our location in the multiverse is random. We can divide this into two steps: First, the universe \(u^{q}\) in which we happen to be is chosen randomly. Second, our \"location\" \(s\) within \(u^{q}\) is chosen at random. We call these the _universal self-sampling assumption_. The crucial difference to the informal anthropic self-sampling assumption used in doomsday arguments is discussed below. Recall the observer program \(\tilde{s}\! :=\!rqs\) introduced in Section 5. We will make the simplifying assumption that \(s\) is the identity, _i.e._, restrict ourselves to \"objective\" observers that observe their universe completely: \(\mbox{UTM}(\tilde{s},\!\ddot{u})=o^{qs}=\mbox{UTM}(s,\!u^{q})=u^{q}=\mbox{UTM }(q)\). Formally, the universal self-sampling assumption can be stated as follows:  
_A priori it is equally likely to be in any of the universes \(u^{q}\) generated by some program \(q\!\in\!\{0,\!1\}^{*}\)._  
To be precise, we consider all programs with length bounded by some constant \(L\), and take the limit \(L\!\rightarrow\!\infty\). **Counting consistent universes. ** Let \(o_{1:t}^{true}\!=\!u_{1:t}^{true}\) be the universe observed so far and  
\[Q_{L}:=\{q:\mbox{Length}(q)\leq L\mbox{ and UTM}(q)=u_{1:t}^{true}*\}\]  
be the set of all consistent universes (which is non-empty for large \(L\)), where * is any continuation of \(u_{1:t}^{true}\). Given \(u_{1:t}^{true}\), we know we are in one of the universes in \(Q_{L}\), which implies by the universal self-sampling assumption a uniform sampling in \(Q_{L}\). Let  
\[q_{min}:=\arg\min_{q}\{\mbox{Length}(q):q\in Q_{L}\}\quad\mbox{and}\quad l:= \mbox{Length}(q_{min})\]  
be the shortest consistent \(q\) and its length, respectively. Adding (unread) \"garbage\" \(g\) after the end of a program \(q\) does not change its behavior, _i.e._, if \(q\!\in\!Q_{L}\), then also \(qg\!\in\!Q_{L}\) provided that \(\mbox{Length}(qg)\!\leq\!L\). Hence for every \(g\) with \(\mbox{Length}(g)\!\leq\!L\!-\!l\), we have \(q_{min}g\!\in\!Q_{L}\). Since there are about \(2^{L-l}\) such \(g\), we have \(|Q_{L}|\!\gtrsim\!2^{L-l}\). It is a deep theorem in algorithmic information theory [6] that there are also not significantly more than \(2^{L-l}\) programs \(q\) equivalent to \(q_{min}\). The proof idea is as follows: One can show that if there are many long equivalent programs, then there must also be a short one. In our case the shortest one is \(q_{min}\), which upper bounds the number of long programs. Together this shows that  
\[|Q_{L}|\;\approx\;2^{L-l}\]  
**Probabilistic prediction. ** Given observations \(u_{1:t}^{true}\) we now determine the probability of being in a universe that continues with \(u_{t+1:n}\), where \(n\!>\!t\).