deletions | additions       diff --git a/untitled.tex b/untitled.tex  index fde35d5..fdcd5c5 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
\section{Disclaimer}  The paper below started as a mathematical attempt to understand some things, what it would mean to live in a world that is not designed,  but, in the end, the mathematical part turned out to be rather small, containing only a few simple properties about set cardinalities and probabilities. I think that the non-mathematical ideas are fairly obvious consequences of the mathematical ones, so probably many people have already thought about them. I tried searching about it and to find this idea on the internet, but  the closestthing that  I found could get  is the idea that the order of the Universe implies or suggests that there is a God. The fine-tuning of the Universe is also close\footnote{\href{https://en.wikipedia.org/wiki/Fine-tuned_Universe}[https://en.wikipedia.org/wiki/Fine-tuned_Universe]}. However, I think that what I'm presenting in this paper is different from what I have read about both of these. I have found quotes from various people that seem to hint at the idea below, but I did not find yet anyone that tried to actually develop it. Since it started as a mathematical paper I may use \ghilimele{we} instead of \ghilimele{I} more often than I should, but you should consider it an invitation to work together in discovering some ideas. And if some of those ideas are wrong or unclear\footnote{given my lack of experience with philosophy this is more probable than I would like}, I welcome counterarguments and feedback\footnote{Authorea allows everyone to comment on the document. I may switch to a different commenting system if it turns out that something better is needed.}. 
...
Many people believe that the world is designed and created and that it's unreasonable to believe that any world can exist without being created, and I agree with them. However, these beliefs are not shared by everyone, so it's worth thinking about what this means. If the world is created, then it's likely to be the way it is because its Creator\footnote{Not everybody that believes that the world is created thinks that God created it. Still, I hope that they would agree that naming the creator of this world as \ghilimele{the Creator} is reasonable.} wanted it to have certain properties. In order to understand why our world works the way it does, one would need to understand the intent of its Creator. While that is interesting in itself, I will not try to pursue it here.  Let us consider the other case: for most of this article, let us assume that our world was not designed and created. If that's true then there may be other worlds\footnote{We don't have any proof for the existence of other worlds, but one could expect them to exist for the same reason that ours exists. If ours has no reason at all for existing, which is likely if it is not created, then it's likely that other worlds would also not need any reason for existing and would simply be. However, for this paper it does not matter if there are other worlds or not and probably we wouldn't be able to tell if other worlds exist or not.}. Even if there are no other worlds, ours one  could have been different. easily imagine a world where, say, the speed of light is different or where gravity works differently.  We will denote by \definitie{possible worlds} these other worlds that either are or could have been. \section{Modelling possible 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, if we relax the countability requirement, 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. We will call these the \ghilimele{too-general} problem and the \ghilimele{too-specific} problem.  For a given world, a good \definitie{good  set of axioms axioms}  would be one that would allow us to make all possible correct predictions for that world. The term prediction is not a clear one. To make it more clear, let us restrict again the \ghilimele{possible worlds} term. One option would be to make it to denote all possible worlds which have a concept of time and a concept of the state of the world at a given time and for which describing the state of the world at all possible times is equivalent to describing the world. 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 \ghilimele{the state of the world at a given time}, so we could rephrase the definition above to include many other reasonable notions of space and time, e.g. we can include worlds where \ghilimele{point} is a concept and we can know which (point, time) pairs are before a given (point, time) pair.  Then, in the following, we will say that we can \definitie{predict} something ($S$) whenever we have a set of axioms for which $S$ is uniquely determined by the state of the world at a subset of the previous points in time\footnote{Will be extended to statistical predictions in the next paragraph}. If we are interested in predicting the state of the world at a given point $P$ and time $t$, a good choice for this subset could be a full section through $P$'s past (e.g. a plane which intersects it's past cone), i.e. a subset that separates $P$'s past in two parts, one which is before the subset and one which is after the subset\footnote{This means that all lines which fully lie in $P$'s past and connect a point which is before the subset with a point which is after the subset must go through the subset}. One could think of similar definitions for predicting the entire state of the world. If needed, this definition could be changed to work for more concepts of space and time.  In a deterministic universe, if we know the laws of the universe and its full state at a given time, we could, in theory, fully predict any future state. But an universe does not have to be deterministic and, even if it is, one could have the  only reasonable model available at a certain time may be  a statistical model for it. one.  Then we will allow using a set of axioms which only gives a statistical distribution for the state of the universe given its past (I'll call this a \definitie{statistical axiom set}). For the purpose of this document we don't need to make a difference between a non-deterministic universe and a deterministic one but for which we only have a statistical model. Let us restrict the possible worlds term to the worlds where we can make predictions and let us use only sets of axioms that allow predictions. As mentioned defined  above, for a given world, a good set of axioms is one which allows us to make all possible correct predictions for that world (statistical or not). Using only good sets of axioms solves the \ghilimele{too-general problem} since such a set would describe its possible worlds in a way that does not leave out important things. Still, there is nothing that prevents such a set from going into too much details. detail.  Let us chose any formalism for specifying axioms that uses a finite alphabet and let us choose an arbitrary order relation on that alphabet. Also, we will only let us  consider only  axiom sets which, when written on an infinite paper, use at most a countable number of symbols. Then, for each possible world, we could say that the best set of axioms is the smallest good one, smallest being defined as having the smallest length when written on (possibly infinite) paper. This is not a well defined notion, so let us also say that for sets of axioms having finite length we will break any ties by using the lexicographic order. For worlds having only sets of axioms of infinite length we can't use the same trick to break ties, but they all have the same length (countably infinite) so we will choose any such set of axioms and say that it's the best one\footnote{Note that we could try a few things to find a canonical set for the infinite case, e.g. whenever a set of axioms is implied by a smaller finite set we could replace it, but it's not clear that any such method will be enough}. With an axiom set chosen in this way we would also solve the \ghilimele{too-specific problem} in the finite case. In the infinite case we [TODO: have?]  made an arbitrary decision, but then the cardinality of the axiom set matches the complexity of the world anyway, even if there is some redundancy. If $U$ is an universe and $A$ is the smallest axiom  set of predictive axioms chosen  asdescribed  above, then we would say than $A$ is the \definitie{optimal set of axioms for $U$}. If $A$ is a set of axioms which is optimal for some universe $U$ then we say that $A$ is an \definitie{optimal set of axioms}. Now let us see if we actually need infinite length sets. We can have infinite sets\footnote{Note that for a given finite alphabet, i.e. in the current context, the \ghilimele{infinite length set} notion is identical to \ghilimele{infinite set}.} of axioms, and there is no good reason to reject such sets and to ignore their possible worlds, so we will take them into account. It is less clear that we can't replace these infinite sets with finite ones. Indeed, let us use any binary encoding allowing us to represent these sets as binary strings, i.e. as binary functions over the set of natural numbers (it's not important to use binary strings, the construction below would work with many other mathematical objects). The encoding of a set of axioms $A$, $encoding(A)$, would be a function from the natural numbers to a binary set, giving the value of the bit for each position in the encoding, $encoding(A):\naturale\longrightarrow \multime{0, 1}$. Then the following scenario becomes plausible: for any universe $U$ with an infinite set of axioms $A$, we can consider $U+encoding(A)$ to be an universe in itself which has $encoding(A)$ as part of its state at any moment in time. Then it's likely that we can find a finite set of axioms which allows predictions for such an universe. While, strictly speaking, this would be a different universe than the one we had at the beginning, it may seem similar enough to it so one may be tempted to use only finite sets of axioms. 
...
Then, for the given intelligent beings we would say in the first case that their universe has a \definitie{finite observable description} and in the second and third case that it has an \definitie{infinite observable description}. Of course, a possible universe $U$ could have multiple types of intelligent beings, each type perceiving the universe in a different way. Because of this difference in perception, for some intelligent beings the universe $U$ may have a finite observable description while for others it may have an infinite observable description.  If we can have infinite descriptions, then the set of optimal systems of axioms\footnote{Using \ghilimele{set of axioms} in some contexts may make the text harder to read, so I'm replacing it with \ghilimele{system of axioms}.} would have the same cardinality as the set of real numbers $\reale$. \footnote{Note that for easier writing, I will sometimes use $\reale$\ as a number, i.e. \ghilimele{the number of is $\reale$} instead of more complex phrases that uses \ghilimele{cardinality}.}  Indeed, for the planets-on-a-line example above, we could select any subset of planets and get an universe with an optimal set of axioms that is distinct from any other subset. The set of all subsets of $\naturale$ has the same cardinality as $\reale$, so the set of optimal sets of axioms would have at least this cardinality. On the other hand, each set of axioms is written using an at most countable number of symbols over a finite alphabet, so there can't be more than $\reale$ sets of axioms. [TODO: Add a footnote or a chapter at the end with more mathematical justification for this. Maybe add there the half-proof given here. Maybe I shouldn't bother.] 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, i.e. the laws of the universe as observed at a given time and place can be quite different from the laws 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). 
...
While this is designed such that we can't directly say anything about a specific observable description, we can say things about what has a real chance of being true for a random observable description. First, let us note that any property that is true for only one description has a zero probability (i.e. it is false virtually everywhere). Even more, any property which is true for a countable number of descriptions has a zero probability. This means that any property with a non-zero probability is for sure true for an uncountable number of descriptions. Of course, there may be properties which are true for an uncountable number of descriptions and still have a zero probability.  Now, it could happen that for any description there is a zero-probability property which is true for that description, making it likely that we can't find anything interesting this way. Fortunately, this is not true. Indeed, a property is written using a finite alphabet and has a finite length, so there is at most a countable number of such properties. Let $Y$ be this set. Then $P(Y)$, the probability of the set of observable descriptions for which at least one proposition in $Y$ is true, is at most  the sum of the probabilities of all elements in $Y$, so $P(Y) = 0$. We can then say that for virtually all descriptions, only properties with non-zero probability are true. This means that, if the probability of our world being designed is non-zero, the only rational choices are that either our world is designed or only non-zero probability properties are true.  Now, let us return to the issue of observable descriptions being finite or infinite. With an finite alphabet, only a countable the  set of finite  observable descriptions are finite. is countable.  Then the \ghilimele{is finite} property is a zero-probability one, so either our universe is designed, or at any point in time there will be an important part of our universe that we can observe but can't model no matter how hard we try. \section{Approximations} 
...
\begin{itemize}   \item Restrict ourselves to universes in which we can measure things with real numbers.  \item Fix $\delta \gt 0$ and say that we care about measuring things which are larger than $\delta$. This means that we can have three sizes $a$, $b$ and $c$ with $a=b$ and $b=c$ but $a\not=c$. This should be fine as long as we're aware that equality here actually means that the difference is smaller than $\delta$.  \item Fix a time length $s$ for the \ghilimele{happens rarely} definition  and ignore things which happen rarely. \end{itemize}  We could actually  use any reasonable definition of measuring and happen rarely. happens rarely, the ones above are provided as an example.  Then we could say that the important things are the ones which are larger than $\delta$ and which do not happen rarely. Let us also fix an arbitrary time length $t\ge 0$, a relative error $\epsilon \ge 0$ and a probability $q\ge 0$ which is the probability of a random prediction to be successful given the previous constraints and let us denote by $f$ with $0 < f \le 1$ the fraction of the world\footnote{As above, everything that can be inferred from the artificial restrictions imposed by this paper to the possible worlds is not considered a part of $f$.} where we can make predictions about what happens after the given time length $t$, with the relative error $\epsilon$ and having a probability $q$ that the prediction is correct\footnote{This could be replaced by \ghilimele{having a probability greater or equal to $q$ that the prediction is correct}, which would also work when having a richer probability distribution for the correctness of the prediction}. Then, if the world is not designed, we have a countable number of finite (observable) descriptions out of a $\reale$ total number of descriptions. Then, for any continuous distribution, the probability of having a finite description with which we can make predictions for a time length of $t$, with a relative error $\epsilon$, with a probability $q$ and for a fraction of the world $f$, is $0$. To have a non-zero probability either $t = 0$ (which means that we are not making any prediction, we are just restating the present), $\epsilon = \infty$ (which means that our predictions have no connection to the reality), $q=0$ (which means that our predictions always fail) or $f=0$. We can discard the first option since then we would have no predictions. We can also discard the second and the third since such a description would not be useful in any way. The only remaining option is that $f=0$; as argued above, a description with $f=0$ can actually make sense. Therefore, with probability $1$, we have $f=0$ and the world has an infinite model. 
...
[TODO: De citit What's So Great About Christianity, Dinesh D'Souza]  [TODO: Put an e-mail address in the footnote in the introduction.]  [TODO: Things to check at the end: my macro is used for \ghilimele{quotes}.]  [TODO: Decide between article and paper.]