Virgil Șerbănuță edited untitled.tex  over 8 years ago

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This is an attempt to reason about why our world is the way it is and what we can reasonably believe about it.  Many people believe that the world is designed and created. However, this belief is 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 "the Creator" \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. If our world was not designed and created, 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 could have been different. We will denote by \definitie{possible worlds} these other worlds that either are or could have been. 

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 \definitie{possible worlds} term will denote only the possible worlds which we could model (including models which need an infinitely long description, but which still follow our rules for reasoning).  This notion of "model" \ghilimele{model}  is not precise enough. Let us restrict the "possible worlds" \ghilimele{possible worlds}  term even 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 all the models of any set of mathematical axioms which is at most countable\footnote{We could also go beyond countable axiom sets, but that would complicate things without any benefit.} and has at least a model are possible 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 nobody designed our world then we have no way of preferring a possible world over another, except that there is one of them in which we live. In other words, if our world is the only one that exists, then any other possible world is as likely to have existed as well. If multiple worlds exist, by picking a random world, ours has a the same chance of being picked as any other.  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, 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 "too-general" \ghilimele{too-general}  problem and the "too-specific" \ghilimele{too-specific}  problem. For a given world, a good set of 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 "possible worlds" \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 "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 "point" is a concept and we can know which (point, time) pairs are before a given (point, time) pair.