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How would a possible world look like? It could have exactly the same fundamental laws as ours, but with the matter organized differently. It could have similar laws, but with different universal constants. It could have different fundamental particles (or whatever the basic building blocks of our universe are, assuming that there are any). Or it could be completely different, i.e. different in all possible ways.  It could be that our logic and reasoning are universal instruments, but it could also be that some of these possible worlds could be beyond what our reasoning can grasp and others could have properties for which our logic is flawed. Aknowledging that, let us see if we can say anything about the possible worlds that we could understand and could model in some way. In the following, the \textbf{possible worlds} term will denote only the possible worlds which we could model (at least theoretically). (including models which need an infinitely long description, but which still follow our rules for reasoning).  This notion of "model" is not precise enough. Let us restrict the "possible worlds" term even more more,  to the possible worlds that we could model mathematically, even if that may leave out some worlds. We will also do the reverse and say that any non-contradictory set of axioms models some possible worlds. worlds\footnote{Even if some of those models seem outlandish, there could be something that follows those rules and that is completely separated from anything else, not interacting with any other universe in any way. I would say that this something would be an universe. If you don't like it, in the following I am going to restrict what I call a possible universe.}.  If there isn't any designer for our world then we have no way of prefering preferring  one over the other, except that there is one of them in which we live. It seems that reasoning about all the possible worlds could be very hard, but maybe we could do something easier, maybe we could reason about the mathematical axioms that model the worlds. Let us consider the set of axioms that define a monoid. All groups are models for this set of axioms, but intuitively a group is something more interesting than a monoid and we should include extra axioms for defining it. On the other hand, we could include all possible axioms for each model (e.g. for each monoid), uniquely identifying it, but again, intuitively this is not a useful way of modeling. For a given world, a much better set of axioms would be one that would allow us to make all possible predictions for that world. We will call this these the "too-specific problem" and  the "too-specific/too-general" "too-general"  problem. The term prediction is not a clear one. To make it more clear, let us restrict again the "possible worlds" term to denote all possible worlds that/which? have a concept of time (which is something reasonably well ordered), that/which have a concept of the state of the world at a given time, for which describing the state of the world at all possible times is equivalent to describing the world, and for which given the state of the world up to a given time $t$ one could find the state of the world at a future time $s > t$. Making predictions would mean finding the state at a future time.  This ignores some important issues like the fact that it's reasonable to have a concept of time without having a well defined concept of "the state of the world at a given time". However, one could rephrase the definition above in a suitable fashion for many concepts of time, e.g. for worlds where "point" is a concept and we can know which pairs (point, time) are before a given (point, time) pair, prediction could mean predicting the state of the world at a given point from the state of the world at previous points in time.  If we define "prediction" in some useful way, as suggested above, and restrict the "possible worlds" term to the ones where we can make predictions, then it makes sense to use only systems of axioms that allow predictions. This solves the "too-general" part of the "too-specific/too-general" problem since such a system would describe its possible worlds in a way that does not leave out important things. Still, there is nothing that prevents such a system from going into too much details. Given a specific formalism for specifying axioms, for each possible world, we could only consider the smallest set of axioms that allow predictions, smallest being defined as "having the smallest length when written on paper". This is not a well defined notion for a few reasons. First, there could be multiple systems with "the smallest length" (one obvious case is given by reordering the axioms). In such a case, we could define an order for the symbols that we are using in our formalism and we could pick the system with the smallest length and that is the smallest in the lexicographic order. Second, there could be systems of axioms of infinite length. For this, we will only consider systems which, when "written on an infinite paper", use a countable set of symbol places on that paper and we will say that all the infinite length systems have the same length, but all of them have a length greater than any finite length. We will ignore systems which need an uncountable set of symbol places. 
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First of all, let us assume that those intelligent beings are continuously trying to find better models for their world and, at any given time, they are trying to use the most simple model which is reasonable at that time (the model could, of course, change as they find out more things about their world) and that they are reasonably efficient at this, i.e. they don't need to find the most simple model, but they aren't very far from it, for a reasonable definition of "very far" (e.g. for each length $n$ of the most simple model there is a maximum length $Max_n$ such that the actual model is not longer than $Max_n$).  Then we would First, let us note that having intelligent beings in an universe likely means that their intelligence is needed to allow them to live in that universe, which likely means that they can  have two possible cases. a partial model of the universe. That model does not have to be precise (it could be made of simple rules like "If I pick fruits then I can eat them.") and it can cover only a small part of their world, but it predicts something. Note that also, their model does not have to be deterministic.  First, those intelligent beings could, at some point in time, find a model which is the best Then we would have three  possible model that they could find for their world. They could stop because they found the perfect model for their world or because the model is so precise that it seems to predict everything that happens in their world and they can't find anything which is not part of their model. We could also relax these conditions by not requiring them to find the best model, but to find one that is good enough for all practical purposes. As an example, for an universe based on real numbers, knowing the axioms precisely with the exception of some constants and measuring all constants with a billion digits precision might (or might not) be good enough. Only caring about things which occur frequently enough could also be good enough. cases.  Second, if First,  those intelligent beings could, at some point in time, find a model which is the best possible model that they  could study find for  their universe forever, world. They could stop because  they would improve found the perfect model for  their models world or because the model is so precise that it seems to predict everything that happens  in their world and they can't find anything which is not part of their model. We could also relax these conditions by not requiring them to find the best model, but to find one that is good enough for all practical purposes. As an example, for an universe based on real numbers, knowing the axioms precisely with the exception of  some essential way forever. constants and measuring all constants with a billion digits precision might (or might not) be good enough. Only caring about things which occur frequently enough could also be "good enough".  For the given Second, those  intelligent beigns we would say in the first case that their universe has beings could reach  a finite observable point where their theory clearly does not fully  modeland in  the second world, but it's also impossible to improve. This could be the  case it has if, e.g., they can model a part of their world, but modelling the reminder would require adding  an infinite observable model. Of course, a possible universe could have multiple types set  of intelligent beings axioms  and some would find a no  finite observable model and others set of axioms  would find an infinite observable get one a better  model. If we can those intelligent beings could study their universe forever, they would improve their models in some essential way forever. Since they  have an  infinite descriptions, then it time, they can and may use strategies like generating possible theories in order, checking if they seem to make sense and testing their predictions agains their world, so if there  is likely a possible improvement to their current theory, they will find it at some point. Note  that in this case  the set fraction  ofdescriptions would have the same cardinality as  the set of real numbers $\reale$. Indeed, one can find an infinite set of disjoint sets of axioms world  thatare different enough from which one  can select any infinite subset and join be modelled (if that notion makes sense) is increasing, but is limited, so it converges at some value. Also,  the subset to form another set of axioms which prediction error [TODO: define]  is predictive, has decreasing and is limited, so it converges. If the fraction converges  at least a model 1  and can't be defined the prediction error converges at 0, then we are  in the first case, because we reach  a finite way [TODO: example point when the fraction is so close to 1 and the error is so close to 0 that one would find them "good enough". If the fraction  or a better explanation. And a demonstration if needed]. the error converge[S?] to different values then we are in the second case.  There is also a third case when there is no reasonable way to define the fraction of the world that can be modelled. As an example, imagine a world with an infinite number of earth-like planets that lie on one line and with humans living on the first one. The planets would be close enough that humans would have no issues travelling between them. Light would come to them in a different way than in our world, but all of them would have enough light. The laws of this hypothetical world, as observed by humans, would be both close enough to the laws in our world so that humans can live on any of the planets, but also different in an easily observable way. Let us say that, starting at 10 meters above ground, gravity would be described with a different function on each planet. On some planets it would follow an inverse of a planet-specific polynomial function of the distance, on others it would follow the inverse of an exponential function, on others it would behave in some way if the distance to the center of the planet is even and in another way if the distance is odd, and so on.  In this case one could study each planet and add a specific description of the laws for each, but at any moment in time the humans in this world would only have a finite part of an infinite set of laws, so we wouldn't be able to say that they cover a non-zero fraction of the laws. If one would think that they cover a non-zero fraction because they cover a non-trivial part of the fundamental forces, then we could also vary the other forces from one planet to the other or we could add other forces. The point is that one can't speak of a fraction of the world that is modelled, even if one is able to model meaningful things (or, at least, the fraction is always 0).  [TODO: Make sure these things work for nondeterministic universes]  For the given intelligent beigns we would say in the first case that their universe has a finite observable model and in the second and third case that it has an infinite observable model. Of course, a possible universe could have multiple types of intelligent beings and some would find a finite observable model and others would find an infinite observable model.  If we can have infinite descriptions, then the set of descriptions would have the same cardinality as the set of real numbers $\reale$. Indeed, we can find an infinite set of disjoint sets of axioms that are different enough and from which one can select any infinite subset and join the subset to form another set of axioms which is predictive, has at least a model and can't be defined in a finite way [TODO: example or a better explanation. And a demonstration if needed].  These observable models of possible worlds are general enough and different enough that it's hard to say something about them, except that they make sense in a mathematical way. Still, given any mathematical proposition $P$ we could try to see what is the chance that it's true in a random observable model. If we can agree on what "true" means, we could ask the same thing about any natural language proposition. [TODO: use the terminology in "on the plurality of worlds"]  In order to compute the probability of a proposition $P$ we would need a statistical distribution for the set of observable models. Unfortunately, we have no good way of chosing among the many possible distributions. Still, there is a class of distributions which stands out as being reasonable. As long as we don't see any reason to prefere a specific description over all the others, our only choice is to use continuous distributions, that is, distributions for which the probability of any given observable model description is zero. 
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We can then say that for virtually all models, only propositions with non-zero probability are true. This means that, if the probability of our world being created is non-zero, the only rational choices are that either our world is created or only non-zero probability properties are true.  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 fix $\epsilon \gt 0$ say that we care about measuring things with a precision $\epsilon$, that we are working only with universes in which we can measure things with real numbers and that we don't care about things which happen rarely, even if we can measure that they happened using the $\epsilon$ precision. 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.  Then, again, our choice is between the world being created and us missing an infinite part of our observable model description which models things that we deem important. If we miss an important infinite part of our observable model description then I argue that we could not make any long term prediction or speak about the relatively distant past because, by definition, the important parts that we are missing would change the outcome of our predictions too much. Therefore saying that our world is (say) 100 years old would have roughly the same chance of being true as saying that is n billion years old [TODO: Thinkdeeply  about this, it's tricky.]  When predicting weather we also can't make long-term precise predictions, and this happens because weather is chaotic??, that is, a small difference in the start state can create large differences over time. However, in an infinite observable model we wouldn't be able to make long term predictions because we don't actually know how the world works, not because we don't know its state precisely enough.  \section{Introduction}  A Christian believes that God created the Universe, but tricky. If  there are many people that think that there is no Creator, implying that the Universe is not created. I think it's worth thinking about what that means and I will try to make a prediction from the fact that there is no Creator.  First of all, let's see if we can say something about a created universe. For a Christian, it should be an universe in which rational beings (humans) can live. These rational intelligent  beingswould be able to study and understand the Universe. In general, however, it depends on who created it. It's likely that it would be optimized for some purpose, but I can't say more without knowing more about what its creator intended.  If our Universe wasn't created, then it should be a random one. I'm going to argue that we can say some interesting things about a random universe.  First, let us note that there may be other universes than ours. Even if there aren't other universes, then ours could have been different. Let us call these other universes \textbf{possible universes}.\footnote{If there is no Creator, I can't help but think that there should be many independent universes, since whatever reason our Universe has for existing, it's very likely that it applies to other universes too. If there is no reason for the existence of our Universe, then the other universes also don't need any reason for existing and they simply exist. Still, the argument I'm trying to make also works when our Universe is the only one that exists.}  Let us define some terms for the reminder of this document.  If a statement $S$ is true  in a possible universe $U$ world  thenwe say that $S$ is a \textbf{property} of $U$.  A \textbf{universe description} $D$ is a set of noncontradictory mathematical axioms that describe the "laws of the universe". I will call each of the mathematical models in which these axioms are true a possible universe for the description $D$ and $D$ is a description for each of this possible universes. Note that a universe will have many possible descriptions and a description may describe multiple universes.  DE PUS MAI INCOLO  Given a universe that has a concept of time, the laws of the universe allow that, given the state of the universe at a time $t$, one can predict the state at any later time.  One could argue that a mathematical definition of the laws of a possible universe is too limiting and that there may be possible universes without this kind of description. That could be, and we could replace "mathematical axioms" with "natural language sentences" without changing much of what follows, except for making the argument somehow more complicated. Still, even "natural language sentences" could be too limiting, but the argument below only needs that our universe has a mathematical description of its laws.  A description $D$ \textbf{implies} a statement $S$ if $S$ is $S$ is a property of all universes $U$ for which $D$ is a description. In this case I will also say that $S$ is \textbf{true for} $D$.  If we have a probability $p$ on a set of universe descriptions $\cal S$ then the probability $p(S)$ of a sentence of a sentence $S$ is the probability of the set of descriptions for which $S$ is true.  Given the definitions above, this is the layout of my argument.  \begin{enumerate}  \item A possible universe description is a set of mathematical axioms. Any set of noncontradictory mathematical axioms which is at most countable is a possible universe description.  \item There is an infinite and uncountable (of cardinality at least $\reale$) set of possible universe descriptions in which Turing-complete entities could exist, which are fundamentally different and for any two such descriptions their sets of possible universes are disjoint. The Turing-complete entities have access to data which could be used to provide a description of the universe and the descriptions which could be built this way also form an uncountable set. [TODO: maybe replace Turing-complete with intelligent and expand on the descriptions provided by the intelligent entities]. Let us denote this set by $\cal T$.  \begin{enumerate}  \item There is such a set which has descriptions somehow similar to our universe, making them more plausible.  \end{enumerate}  \item For any sentence $S$ which is true in a countable set of universe descriptions and any probability distribution which is not heavily biased (i.e. any continous probability distribution) the probability of $S$ is zero.  \begin{enumerate}  \item Given such a probability distribution over $\cal T$, let $\cal T_0$ be the set of all descriptions in $\cal T$ in which there a true sentence $S$ of probability zero. Then the probability of $\cal T_0$ is zero. Note that the sentence is not of the type mentioned above.  \item There set of finite descriptions is countable, so the property "Has a finite description" has a probability 0 in $T$. [TODO: This construction should also work when the descriptions are not disjoint, but they allow full predictions of events in  their universes].  \end{enumerate}  \item An intelligent being inside an universe $U$ with an infinite description $D$ in $\cal T$, trying to find a description of $U$ can notice that the universe description becomes more and more complex with each observation, making an infinite description likely.  [TODO: This in unnatural. As defined here, the universe description becomes more complex only inside $\cal T$. The universe description must allow full predictions.]  \end{enumerate}  znzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz  Is there anything that we could say about an uncreated universe? Psychologically it's hard to move away from what we know about this world, many people would answer something similar to "It would be very similar to our Universe". However, it is clear that this view is rather limited and, with some effort, one could imagine universes which have nothing in common with ours. To avoid being blinded by what we currently know, let's try to imagine that we are rational beings that, somehow, don't have any knowledge about any universe and that can't observe anything around them and let's think about how an universe could be like. We may think that an universe could be something completely opaque to rationality, that we couldn't describe any laws that govern it and, in general, that we could say only trivial things about it (e.g. "It's an universe", "I don't understand it").   Although this may be reasonable, it may be more useful to think about universes which we can understand rationally. I would say that one can understand an universe in a rational way if and only if that universe can be modelled mathematically. I think that the reverse is also true, if something can be modelled mathematically, it could be an universe. If this seems too outrageous, you should know that this document works mostly with more plausible universes. However, I don't see a good reason to be outraged by having, say, $\intregi_2$ as a possible universe.  Anyway, an universe without rational beings is not that interesting, because nobody would know that such an universe exists. One of the most simple universes in which something rational may exist is a Turing machine. It is not clear if such a machine would be enough for having rationality, but it can produce every mathematical result that we could discover, which is impressing enough. I guess that this would be a solipsistic universe.  Let's focus on a restricted set of universes. Let's say that all have a 3d space with real coordinates and there is something that we call "time" and which contain some fundamental objects which are functions $f:\reale^3\longrightarrow\complexe$). Let's say that these functions are similar to the wave functions used by quantum mechanics. "Similar" is rather vague but should be good enough for what follows.  Let's say that the fundamental objects of each of these universes can be grouped in "types" (electrons and photons could be such types) and we may have a possibly infinite set of types of objects. Let's assume that form of the interactions between fundamental objects is given by their types. As above, these rules would be similar to the ones in quantum mechanics. I think that these restricted universes would be reasonable even for people that find $\intregi_2$ to be too strange.  Let's try to make the "type" and "interaction" concepts more clear. At a given time, a concrete object is defined by its function, but also by its "type". Knowing all objects in the universe at a given time $t$ with their types and functions and using the rules that describe the interactions between types we should able to predict the objects at any time $t_1 > t$.  This is an intuitive explanation, one wouldn't be able to say what in included in a type and what isn't. As an example, although "foton" and "electron" seem like sensible types and that the concrete objects are wave functions, we could actually say that we have only one type, "foton or electron" and a concrete object would be, say, a function $f:\reale^3->\complexe \times \{\mbox{foton, electron}\}$, asking that the last component of $f(x_1, x_2, x_3)$ is a constant for any given function $f$. This function is not of the form required above, but one could think of several tricks to transform it into such a function.  Also, instead of joining several types into one, we could have an infinite number of types like „foton having energy $x$”.  I will not fully solve this ambiguity, but I will restrict it. First, I will request that a type's description consists of a finite number of sentences. Also, if the laws of an universe (types and interactions) can be described in more than one way, I'll use the shortest description (with the lowest cardinality for the ordered set of words that describe it).  One issue is that such a description may need some fundamental constants that are real numbers which are impossible to describe in a finite way. To solve this issue, we will use only constants that have a finite description. We would not be able to tell apart a real number from a very good finite approximation by a rational number. If one is really concerned about this, we could change the condition about being able to predict the objects at a future time above with a condition in which we are able to approximate the objects after a given time. \footnote{Also, it's not easy to see, but by thinking about the construction below, this works even when using only rational numbers.}  \section{Some statistics}  For any continuous distribution, any countable set has a zero probability. This means that we can safely bet that we will not chose any number out of that set. \footnote{See e.g. http://www.math.uah.edu/stat/dist/Continuous.html}.  As an example, if the time interval between two consecutive particles produced by radioactive decay is indeed a random real number then we can be sure that it can't be a whole number of seconds.  But, since the set of rational numbers is countable, we can be sure that the time interval can't be a rațional number of seconds (even if in real life we would not be able to distinguish it from one). If we try to extend the set of rational numbers by using all arithmetic operations and roots we would still get a countable set of numbers. Even if we include all algebraic numbers (roots of equations of type $P(x)=0$ where $P$ is a polynomial with rational coefficients), we still get a countable set. Even more, the set of all real numbers that we can define using a finite number of words (which contains transcedental numbers like $\pi$ and $e$) is countable.  In other words, we can be sure that the length in seconds of the time interval can't have a finite definition. Note that we can still say some things which have a nonzero chance of being true, like "The first digit after the decimal point is 2".  \section{A mathematical-ish approach for descriptions of universes}  Although a large part of this section is mathematical, there are non-mathematical parts. For this reason I used "Argument" instead of "Proof". The main issue is whether the arguments are solid enough from a rational perspective, not if they are fully mathematical. Hopefully in the future I'll extract the mathematical part in a separate section.  \begin{definitie} A \textbf{description of a possible universe} consists of a finite set of axioms which describe the base sets ($\reale$, $\complexe$, etc.) and an set of axioms which can be infinite and which describe:  \begin{enumerate}  \item object types;  \item the wave functions for each type of objects;  \item rules for interactions between objects of given types.  \end{enumerate}  For each type, the set of axioms that describe it and its wave functions should intelligence is likely to  be finite. For each two types, the set of axioms useful. Then it means  that describe the interactions between at least a part of  the two types should world must  be finite. We understandable and predictions  canhave an infinite set of types. We will assume that the set of types is at most countable, but we could also work without this assumption. We will also assume that the full set of axioms is not contradictory.  \end {definitie}  If one finds fully mathematical descriptions to  be limiting, we could start our descriptions made  withsimple concepts that an average human would understand and build using that as a foundation. Of course, such  a description could include axiomatic definitions of, say, various branches finite model  ofmathematics. However, I think that these descriptions would not change  the validity of world. But then  the constructions below, but model they  would still make everything less precise.  As hinted above, another issue build  is infinite. This means  that they can find  a possible universe may contain things which are impossible to describe. As an example, most Christians believe that something indescribable exists, although it may exist outside of our Universe. Still, model  that would not change anything in the reminder allows some sort  of the document.  \begin{definitie}  A \textbf{possible universe} predictions  for a given description $D$ is a set fraction $f$  of pairs of wave functions their world  and types for each moment in time. These wave functions conform to the interaction rules between the types found in $D$. To keep the reminder of the document simple we they can increase this fraction by research. They  will ignore that time instances for different places in space may not be totally ordered.  \end{definitie}  \begin{definitie}  Two entity types are connected if their objects may interact.  Două tipuri de entități sunt \textbf{conectate} dacă interacționează între ele.  \end{definitie}  \begin{definitie}  Two descriptions are \textbf{equivalent} never have $f=1$, but  if there is an isomophism between their universe sets [TODO: De spus ce e un universe set pentru o descriere, de spus ce e un izomorfism].  \end{definitie}  \begin{afirmatie}  If $M$ is the set of universe descriptions then $M$ has the same cardinality as $\reale$.  \end{afirmatie}  \begin{argument}  Let $A$ be the set of all sentences. $A$ is countable. $M$ is included in the power set of $A$ so we denote  the cardinal of $M$ is theory they have  atmost $\reale$.  There is  a countable set time $t$ by $T_t$ and the fraction  of sentences $P = \{a=a, b=b, \dots\}$ which the world that they  can be added and removed predict by $f_t$ then $f_t$ converges  to $1$ when $t$ converges to infinity. But then for  any description in $M$ without actually changing it, so the cardinal fraction  of $M$ the world $f<1$, no matter how close to $1$, there  is a finite time  atleast the cardinal of the power set of $P$,  which has the same cardinal as $\reale$.  \end{argument}  However, this ??? does not tell us much because, as it's obvious from its proof, any set of universes there is a theory  that has can predict  a description has an uncountably infinite number of equivalent descriptions. However, the next ??? tells us something more interesting.  \begin{afirmatie}  Let $L$ be the set of equivalence classes fraction $f$  ofuniverse descriptions. Then $L$ has  the same cardinal as $\reale$.  \end{afirmatie}  \begin{argument}  În acest argument, dacă $E_1, E_2, E_3, E_4$ sunt niște tipuri de entități, nu neapărat distincte, iar $I_{1,2}, I_{3,4}$ sunt regulile de interacțiune între $E_1, E_2$ respectiv $E_3, E_4$, atunci spunem că $I_{1,2}$ și $I_{3,4}$ sunt \textbf{neechivalente} dacă descrierea formată din $E_1, E_2, I_{1,2}$ nu este echivalentă cu descrierea formată din $E_3, E_4, I_{3,4}$. Putem cere ceva mai puternic decât neechivalența, spre exemplu $I_{1,2}$ și $I_{3,4}$ ar putea fi bazate pe polinoame de grade diferite.  Evident, există un șir infinit de tipuri de entități $E_1, E_2, \dots$ și reguli de interațiune $I_{ij}$, toate neechivalente (sau „puternic neechivalente”, ca mai sus), astfel încât pentru orice $k$, $E_k$ interacționează cu mulțimile precedente $E_1, \dots, E_{k-1}$, iar $I_{ij}$ descrie cum interacționează $E_i$ cu $E_j$. Ca o paranteză, aceasta înseamnă că interacțiunile se pot descrie pentru perechi de $E_i$-uri și nu e nevoie să considerăm triplete, de exemplu.  Putem alege aceste $E$-uri și $I$-uri astfel încât sunt independente în sensul independenței unor axiome, adică, de exemplu, nicio interacțiune $I_{ij}$ și nici negata ei nu se pot deduce din cele precedente.  Atunci orice submulțime de tipuri de entități împreună cu regulile de interacțiune care li se aplică descrie o clasă de universuri diferită de celelalte și se află într-o clasă de echivalență diferită.  Mulțimea submulțimilor unei mulțimi numărabile este în bijecție cu $\reale$, deci mulțimea claselor de echivalență ale universurilor posibile este în bijecție cu $\reale$.  \end{argument}  \begin{notatie}  Notație. Fie $F$ mulțimea descrierilor necontradictorii finite de universuri.  \end{notatie}  \begin{afirmatie}  $F$ este cel mult numărabilă.  \end{afirmatie}  \begin{argument}  Mulțimea literelor unui limbaj este finită. Mulțimea secvențelor de litere de lungime cel mult $k$ este finită. Mulțimea secvențelor de litere de lungime finită este reuniunea după $k$ a mulțimilor de secvențe de lugime cel mult $k$, deci este numărabilă. Mulțimea cuvintelor este o submulțime a acesteia, deci este cel mult numărabilă.   Mulțimea secvențelor de cuvinte de lungime cel mult $k$ este cel mult numărabilă. Mulțimea secvențelor de cuvinte de lungime finită este o reuniune de mulțimi cel mult numărabilă, deci este numărabilă. O descriere finită este o secvență de lugime finită de cuvinte, deci mulțimea descrierilor finite este cel mult numărabilă. Mulțimea descrierilor necontradictorii finite este o submulțime a acesteia, deci este cel mult numărabilă.  \end{argument}  \begin{afirmatie}  Pentru orice distribuție statistică care nu favorizează excesiv descrieri de universuri (adică e continuă), probabilitatea lui $F$ este zero.  \end{afirmatie}  \begin{argument}  Fie $d$ o distribuție continuă pe $\reale$ (de exemplu o distribuție uniformă). Fie $F_R$ imaginea lui $F$ în $\reale$ printr-o bijecție oarecare. $F_R$ este numărabilă. Deoarece $d$ este continuă, probabilitatea oricărei mulțimi numărabile este zero.  \end{argument}    \begin{afirmatie}  Într-un univers posibil întâmplător, ales fără a favoriza excesiv vreun univers, fiecare tip este conectat de un număr infinit de alte tipuri.  \end{afirmatie}  \begin{argument}  Fie o mulțime de tipuri $T$. Cu probabilitate $1$, $T$ este numărabilă.  Fie $t$ un tip din $T$. Fie $A_t$ mulțimea tuturor mulțimilor de tipuri cu care ar putea interacționa $t$ într-o descriere de univers oarecare (mulțimea $X$ este în $A_t$ dacă și numai dacă există o descriere de univers care folosește tipurile $T$ și în care $t$ interacționează cu toate tipurile din $X$ și nu interacționează cu niciun alt tip). Evident, $A_t$ este mulțimea tuturor submulțimilor lui $T$ (aceste submulțimi îl pot conține pe $t$ pentru că elementele de același tip pot interacționa între ele), deci se poate pune în bijecție cu $\reale$.  Fie $B_t$ mulțimea tuturor elementelor finite ale lui $A_t$. Atunci $B_t$ este numărabilă, deci, ca mai sus, probabilitatea lui $B_t$ este zero.  Fie $t_1, t_2, \dots$ tipurile din $T$. Fie $Fin(t)$ propoziția „$t$ este legat direct de un număr finit de noduri”. Atunci  $$P(Fin(t_1) \vee Fin(t_2) \vee Fin(t_3) \vee \dots) \le P(Fin(t_1)) + P(Fin(t_2)) + \dots = P(B_{t_1}) + P(B_{t_2}) + \dots = 0$$  Ca urmare, probabilitatea ca într-un univers să existe un tip conectat cu un număr finit de alte tipuri este zero.  \end{argument}  Este posibil ca unele dintre interacțiunile grafulului tipurilor să fie foarte slabe. Să definim (doar la nivel intuitiv) \textbf{puterea} unei interacțiuni între două tipuri $t_1$ și $t_2$ $Pint(t_1, t_2)$ ca fiind cât de mult schimbă o interacțiune cu un obiect de tip $t_2$ parametrii observabili ai lui $t_1$ și să presupunem că o putem măsura cu un număr real pozitiv, astfel încât interacțiunile cu putere mai mare decât un $Pmin(t_1)$ sunt observabile.  Există mai multe moduri de a defini o astfel de putere a interacțiunilor între două obiecte $o_1$ și $o_2$ concrete de tipuri $t_1$ respectiv $t_2$. Pentru a simplifica exprimarea, să presupunem că putem observa/măsura parametrii $p_1, p_2, \dots$ ai obiectelor de tip $t_1$ (de exemplu energia sau poziția) și să presupunem că această măsurătoare o facem cu o precizie $pr(t_1, p_i)$ (valoare mică înseamnă precizie mare, zero ar însemna că putem măsura valoarea exactă; presupunem că nu putem măsura valorile exacte).  Atunci putem lua ca putere a interacțiunii între $o_1$ și $o_2$ maximul raportului între cât se modifică parametrii $p_i$ și precizia cu care îi putem măsura. Dacă $a_i$ este valoarea parametrului $p_i$ înainte de interacțiune iar $b_i$ este valoarea după interacțiune, atunci puterea interacțiunii între $o_1$ și $o_2$ este $max_i(\frac{\mid b_i - a_i\mid}{pr(t_1, p_i)})$.  Să observăm că, folosind această construcție, $Pmin(t) = 1$, unde $t$ este un tip pentru care putem măsura cel puțin un parametru.  Pentru că interacțiunile între obiectele de două tipuri $t_1$ și $t_2$ probabil că nu au o putere constantă, ci puterea variază în funcție de obiectele concrete, definim puterea interacțiunilor între tipuri în funcție de interacțiunile între obiecte concrete. Fie următoarele:  \begin{enumerate}  \item $Pc(t_1, t_2)$ mulțimea puterii interacțiunilor posibile între obiecte concrete de tip $t_1$ și $t_2$.  \item Alegem o distribuție continuă oarecare pe $Pc$.  \item Alegem o probabilitate $p>0$ astfel încât să considerăm că $p$ este limita până la care avem șanse să observăm evenimente care se întâmplă cu probabilitatea $p$ în cadrul unei observații asupra tipului $t_1$.  \end{enumerate}  Atunci luăm $Pint(t_1, t_2)$ ca fiind cea mai mare valoare pentru care avem șanse să observăm interacțiuni, adică  $$P(\{x\in Pc(t_1, t_2)\mid x\ge Pint(t_1, t_2)\}) = p.$$  \begin{afirmatie}  Dacă există două tipuri $t_1, t_2$ cu o putere a interacțiunii suficient de mare pentru a putea fi observată, atunci $t_1$ este conectat cu o infinitate de alte tipuri astfel încât puterea interacțiunii lui $t_1$ cu acestea este suficient de puternică pentru a fi observată. Altfel spus, dacă există $t_1$ și $t_2$ cu $Pint(t_1, t_2) \ge Pmin(t_1) = 1$ atunci există o infinitate de tipuri $s_1, s_2, \dots$ astfel încât $Pint(t_1, s_i)\ge 1$.  \end{afirmatie}  \begin{argument}  Fie $t_1$, $t_2$ cu $Pint(t_1, t_2) >= 1$, deci ale căror interacțiuni pot fi observate.  Ca mai sus, presupunem o distribuție continuă pentru valorile $Pint(a, b)$. Fie $c = Pint(t_1, t_2)$. Atunci pentru orice $\epsilon > 0$ avem $p(Pint(t_1, x)\in \lbrack c, c+\epsilon)) > 0$. Atunci, deoarece avem un număr infinit de $x$-uri, un număr infinit dintre ele vor avea $Pint(t_1, x)\in \lbrack c, c+\epsilon))$, deci un număr infinit vor fi „mai detectabile” ca interacțiunea între $t_1$ și $t_2$. Cu alte cuvinte,   $$p(t\mbox{ are un număr finit de interacțiuni cu }Pint(t_1, x)\in \lbrack c, c+\epsilon)) = 0.$$  Ca mai sus, pentru o mulțime de tipuri și muchii date,   $$p(\mbox{există un }t\mbox{ cu număr finit de interacțiuni observabile}) \le \sum_{t}p(t\mbox{ are un număr finit de interacțiuni observabile}) = 0.$$  \end{argument}  \begin{afirmatie}  Ceea ce știm despre universul nostru indică faptul că a fost creat.  \end{afirmatie}  \begin{argument}  Mai jos $Df$ înseamnă „descriere finită a tipurilor de obiecte și interacțiunilor ce pot fi observate” iar $Ui$ = „univers întâmplător”.  $$p(Ui \mid Df) =\frac{p(Df \mid Ui)p(Ui)}{p(Df)}=\frac{0}{p(Df)}$$  Universul nostru pare să aibă o descriere finită a legilor, descriere dată de mecanica cuantică și teoria relativității, care par a descrie extrem de bine realitatea. O eventuală teorie unificată pare că ar fi tot finită. Din păcate acest lucru nu poate fi demonstrat, putem doar observa rezultatul a sute de ani de cercetare științifică și faptul că de ceva timp încoace nu am mai descoperit nicio lege fundamentală nouă [TODO: de investigat].  Atunci, deoarece avem universul nostru ca exemplu, este rezonabil să presupunem că $p(Df) > 0$, și atunci putem spune că $P(Ui | Df) = 0$.  Cu alte cuvinte, ceea ce putem observa ne indică un univers creat.  \end{argument}  \section{Obiecții}  \subsection{Nu observăm componentele fundamentale}  O obiecție ar fi că noi nu putem observa componentele fundamentale ale universului, ci îl observăm la un nivel mult mai înalt. Însă, intuitiv, dacă universul nu ar avea o descriere finită a legilor, compexitatea infinită s-ar vedea la orice nivel.  \subsection{Proprietatea „descriere finită” e arbitrară}  Chiar dacă probabilitatea ca universul să aibă o descriere infinită și o complexitate observabilă infinită este $1$, oare nu ar putea fi adevărat că pentru orice univers $U$ găsim o proprietate $p$, nu neapărat cea a descrierii finite, astfel încât toate universurile cu proprietatea $p$ au probabilitate zero?  Fie $Prop$ mulțimea tuturor proprietăților pe care le putem exprima ca mai sus, de lungime finită, fie ca propoziții matematice, fie pornind de la concepte care pot fi înțelese de un om obișnuit. Evident, $Prop$ este numărabilă. Fie $Prop_0$ submulțimea lui $Prop$ formată din proprietățile $p$ pentru care probabilitatea universurilor care respectă $p$ este zero. $Prop_0$ este cel mult numărabilă.  Fie $p_1, p_2, \dots$ proprietățile din $Prop_0$.  $$p(U\mbox{ – univers posibil care are una din proprietățile din }Prop_0) = $$  $$p(U\mbox{ are proprietatea }p_1\mbox{ sau }U\mbox{ are proprietatea }p_2\mbox{ sau }\dots) \le $$  $$\sum_i p(U\mbox{ are proprietatea }p_i) = 0$$  În concluzie, toate universurile posibile afară de o submulțime neglijabilă (de probabilitate zero) respectă doar proprietăți care au probabilitate nenulă. universe. ].  \subsection{Creator creat} There is another distinction that we should make. When predicting weather we can't make long-term precise predictions, and this happens because weather is chaotic??, that is, a small difference in the start state can create large differences over time. This could happen even if the universe is determinist and we know the laws of the universe perfectly, as long as we don't know the full current state of the universe. On the other hand, with an infinite observable model we wouldn't be able to make long term predictions because we don't actually know how the world works, not because we don't know its state precisely enough.  Acest argument nu arată că Dumnezeu există, ci doar că lumea noastră este creată. Totuși, un ateu ar putea întreba dacă același argument nu se poate aplica lui Dumnezeu. And there is yet another difference that we should make. In a deterministic universe, knowing the laws of the universe and its full state we could, in theory, fully predict its future. But an universe does not have to be deterministic. In this case, we would search for the best set of laws for predicting the future state. Note that we could have an universe that seems non-deterministic for any finite set of laws but which has an infinite set of laws under which it is deterministic. Indeed, it seems that in a case in which we can easily observe the effects of some laws of the universe, we could probably infer a finite statistical law about it. In this case, the best set of laws would be the infinite one. [TODO: Think about finite statistics. Is it always true? Probably not, if the stats made in a day are completely different from stats made in another day. How frequent would it be? What does it mean?]  Eu susțin că nu se poate, pentru că nu putem face nicio afirmație suficient de precisă despre Dumnezeu. De exemplu putem spune că Dumnezeu este bun și, deși acest lucru este adevărat, conceptul nostru de „bun” este extrem de limitat, și Dumnezeu depășește incomparabil de mult tot ce am putea înțelege noi prin „bun”. Cu alte cuvinte, spre deosebire de descrierile de universuri, nu putem avea descrieri aproximative succesive care să se apropie arbitrar de mult de o descriere exactă. În plus, deși putem spune câteva lucruri despre Dumnezeu, cum ar fi că este bun, există un număr infinit de lucruri pe care nu le știm despre El.    \subsection{Altele} [TODO: Give examples in which our main assumptions about the universe, i.e. homogeneity and isotropy, are broken. Are these finite properties, or zero-probability ones? They are not finite, but considering that we can combine any at most countable set of homogenous and isotropic universes with compatible times into another universe, then it's likely that they are zero-probability ones. We need intelligent beings to be able to live through these changes, but even then it looks like we can combine a lot of universes into one, suggesting that these properties are zero-probability for many reasonable probability distributions. TODO: Give examples on how to combine. Say in a clear way what do I mean by combining a lot of universes into one, making it obvious why the probability should be zero. We experience gravity differently at various times and places - tides, variation from one place to another on Earth, on the Moon, when falling, although the law that describes gravitation does not change. We could imagine an universe where the actual law changes.]  Sunt și alte obiecții, încă nescrise aici. When talking about a mathematical description of the universe as one sees it, it is obvious that the description may depend both on time and place of the observers (assuming that the universe has a concept of place that is close enough to ours). The laws of the universe as observed at a given time and place can be quite different from the laws that one can observe at another time and/or place. If these differences are unpredictable, then an intelligent being will never be able to find a full mathematical description of the universe, even if we assume that it could live through all these changes (as time passes, and/or as it moves through the space). Note that these beings must be able to live through the changes, otherwise the universe does not count for our problem.