What are Sets?
in [Logic]
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From: zuhair on 30 May 2010 14:44 A Set is a collectible object uniquely representing the end result of a collective attempt. A collective attempt refers to the process of collecting objects, which involves various specifics of selecting objects and gathering them together. Two collective attempts might end up in having the same objects being collected, thus they would have the same end results; or might end up in having some objects collected by one collective attempt that are not collected by the other collective attempt, then the end result of the two collective attempts would be different; some collective attempts fails completely to collect any object, and thus the end result of this collective attempt would be designated as failed, that is to discriminate it from successful collective attempt. What we desire is having a criterion that differentiate the various *end results* of collective attempts, such that any two failed collective attempts would be designated by the same symbol; also a failed collective attempt would be designated by a symbol that is different from any symbol given to a successful collective attempt; and further we need to have different symbols differentiating each two successful collective attempts if and only if these ended up with different results; further each end result of a collective attempt should be designated by one symbol. We further desire that those differentiating symbols described above would stand for *collectible objects*. So for example lets take the failed end result of a collective attempt, this would be uniquely represented by an object, now this object would have a differential representational rule as described above, but what we desire also is that this representing object itself can be collected into a collection! i.e. be a collectible object. So what we said above would mount to what is called unique representation of the end results of collective attempts by collectible objects. So those objects that are collectible and uniquely representing the end results of collective attempts are what we call as: *Sets*. Now if we remove the condition of being a collectible object, then those objects that are not necessarily collectible yet uniquely representing end results of collective attempts, are what we call as: "Classes". So while every set is a class, classes on the other hand might be sets, or might not be sets. Classes that are not sets are called proper classes. Now membership can be understood in the following manner. We say that x is a member of y, if and only if, there is a collective attempt T, with the end result R(T) were y uniquely represent R(T) and x is an object collected by the collective attempt T. Definable classes:- Classes are sometimes definable by a property that holds solely for all its members, so all its members have this property while non members do not. In this case we designate that as X={y | Q} were X is the class of *all* objects y fulfilling property Q. Those definable classes are generally viewed as being *object extensions* of the the defining properties, as if the defining property (which is not an object) is personified uniquely into an object in the object world. However we need a group of rules to determine which properties can define sets consistently. And this group of rules is called a Set Theory. Now sets can be in themselves, i.e. they can be among the collectible objects collected by a collective attempt the end result of which this set stands for. That happens when the defining property of that set holds for that set itself; for example the property x=x , this would definitely hold for the set itself since every object is identical to itself. Other times sets might not be in themselves, like the set uniquely representing the end result of a collective attempt that ended up in solely collecting the empty set (the object uniquely representing the failed end result), so obviously this object cannot be a member of itself. This account affords an informal background that justifies the existence of the empty set, also can explain why singletons are not necessarily members of themselves. It also explains non well founded sets, and the difference between a set and a class. So the notion of *set* , *class* ,after all can have an informal meaning, so is the membership relation. In this account I withdraw my previous explanation of sets as containers. Zuhair |