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From: zuhair on 10 Feb 2010 23:31 What are sets? Can we have a simple intuitive understanding of these objects called as sets by mathematicians, like those objects in Z set theory or Aczel's, etc.. Here is a simple trial on my side to understand those rather vague objects intuitively. First lets intuitively interpret sets as *collections*. THE COLLECTION INTUITIVE LINE OF THOUGHT:- Collections arises most clearly when we come across *multiple* objects. *collections* as a concept arises when we deal with multiple objects as ONE entity, and by then we describe the relationship between those objects and this one entity as those objects being members of that entity, from that line of imagination we can then extend the concept of collections to cover single objects, but can we explain the empty set? The above was the context in which the term *collection* is mostly used in common language. Sets are often thought as collections, the term collection is an informal term, it is not well defined. Let's try to go to our world and take simple examples of what we may call as collections, let's examine how they are build and give relevant terminology. "To collect an object" , frequently refers to, "putting an object into a container" Matters start usually by an attempt to collect objects, or what we may refer to as "collective attempt". Now the collective attempt is a term that describes the process of all trials to collect objects under certain given conditions and circumstances, so according to the understanding here it would be a term that describes the process of all trials to put objects into a container as specified by conditions of the attempt. Now collective attempts vary according to the condition required for collecting objects, for example a collective attempt to collect money, is different from that one that aims at collecting a special kind of sea food. Not only that, collective attempts also vary in many other details, however two collective attempts might result in the same objects being collected. What we care about here is the "end result" of the collective attempt, and here we don't care about the various details that lead to these end results. Now we have two main end results of collective attempts. A failed collective attempt A successful collective attempt The failed collective attempt, is a collective attempt that didn't result in any object being collected, i.e. no object is put in the container, so the container into which objects are to be put, remained empty. A successful collective attempt is one that succeeded in putting at least one object into the container. Now we come to define what we mean by the term "collection". ---------------------------------------------------------------------- A collection is the whole of all collected objects. ---------------------------------------------------------------------- If there is no collected objects, then there is no whole of them, then there is no collection. A collection results from a successful collective attempt only! A failed collective attempt results in "no collection" inside the container. So there no empty collection! If the collective attempt resulted in only One object being collected, then the whole of all collected objects would be that object itself! So a collection of one object is that object. If we want to make a formal system about *collections*, then it would be something like the following: let's denote the membership relation by the symbol "e", then we'll have the following formal system in FOL with identity and e. (1)Two collections having the same elements are identical. (2)The empty collection do not exist. (3)The singleton collection is identical to its element. (4)Every element of a collection is singleton. (5)For any formula Phi in which x is not free, then all closures of For all y ( Phi(y) -> y is a singleton) and Exist y Phi(y) -> Exist x for all y ( y e x iff Phi(y) ) are axioms. Which is clearly quite different from the formal systems of Sets like Zermelo's set theories and others. So sets are not collections in the above sense. Now I come to explain what I mean by a "collection status" A collection status is the end result of the collective attempt. More specifically a *collection status* is a term that describes the end result of collective attempt in a differential manner, so if the collection status of a failed collective attempt is named as "K" and the collection status of a collective attempt that succeeded in collecting the object B is to be named as "H", then K must not be identical to H. So two collection statuses are identical if and only if the end result of the collective attempts is the same, i.e. either it is *failure* for both of the collective attempts, or both are successful resulting in the same collection. Accordingly any two collection statuses would be different if and only if one failed while the other succeeded or both succeeded but resulting in two different collections. Now is the collection status an object? The collection status looks as a situation, however this can be looked as an object although definitely of a nature that is quite different from observed objects in our world. So we can understand sets as collection statuses. So: A set is a collection status. So sets are not collections, sets are collection statuses! Now we come to explain what we mean by the "membership" relation: A is said to be a member of X, if and only if, X is the collection status that describes the end result of a collective attempt having A successfully collected (put into the container). The above line of intuitive thinking as to the nature of sets and their members would allow the existence of an empty set, and of sets being members of sets, and of building up higher hierarchy of sets.it serves a strong intuitive basis for simple type theory, and well founded set theories. However the above line of thinking cannot easily explain non well founded sets, the reason is our intuitive habit of imagining the collection status to be formed *after* the end of the collective attempt. Accordingly a set cannot be a member of itself, since this would imply that it started to be formed after it was already formed, which is ridiculous. Another source of difficulty is our habit of imaging sets of physical objects, like sets of apples, of humans, etc.., now the objects in these sets are definitely different in nature from "collection statuses", those objects are "observed entities", while "collection statuses" are descriptive entities, the later are concepts rather than physical objects, and if we are to regard collection statuses as objects, then they are definitely of a very different nature than those observed entities, this lead to the impression that sets are of different kind from their members, and so a set cannot be in itself or in any of the members of its transitive closure, since this entail that it would have a nature that is different from its nature, which is ridiculous. However if we imagine that we are working with pure sets, i.e. sets in which every member of their transitive closures are sets, then all these sets are of the same nature, all of them are "collection statuses", so the difficulty that arise with sets of physical objects is not longer a problem. Also there is nothing to imply the existence of a time factor, A collection status is a *description* of the end result of the collective attempt, but one can imagine that, the collection status and the objects collected all are timeless objects, since both the collected objects and the collection status all are descriptions then there is no real reason for regarding the existence of any time factor in the sense that these descriptions are *formed* in time, and that there is a time lag between them, all these sets are concepts, they are objects yes, but they are conceptual objects, and not a visual-temporal objects like that of the real world. Seeing that pure sets are of the same nature, and that there is no time factor involved, then there is nothing against the existence of non well founded sets. So the above was the collection status line of thought. So the main result is that: ---------------------------------------- A set is a collection status. ---------------------------------------- Now I'll introduce another quite different approach than the above, which is the container line of thought. THE CONTAINER INTUITIVE LINE OF THOUGHT:- It appears that the easier way to explain sets including non well founded sets intuitively is to think of sets as *containers*. Sets can be defined as *content specific containers*, This can be defined formally as: x is a set iff for all y ( for all z ( z e x iff z e y ) -> x=y ) so if x is a container such that there do not exist any other container having exactly the same contents as x, then x is a set, and vise versa, i.e. if x is a set then x is a container such that there do not exist any other container having exactly the same contents as x. While on the other hand distinct containers having the same contents are not sets, they can be called as *para-sets*, but in any case they are not sets. If we imagine sets according to the above intuitive line, then it is not difficult at all to imagine a container that is in itself (a spiral), or in one of its members, or in one of the members of its transitive closure, also it is not difficult to imagine a never ending chain of containers one in each other, all these are possible to imagine. We can even define containment in a visual-temporal way, so A contained in B, which is A e B , if and only if, all parts of A are surrounded by parts of B at some moment of time. So if we imagine A as infinite spiral, then every part of A would be surrounded by a part of A, actually all moments of time, so A would be in itself. We can also follow another line of imagination, though more dynamic, in which we can postulate a time period on membership within which a set can get smaller, and thus be surrounded by itself in the previous time (no longer than the time period of membership) .So the membership relation definition must be adjusted to accommodate such a line of thought. However I prefer the spiral line of imagination. In cases were we have A e B and B e A and ~ B = A, there is no difficulty imaging that also! since there is no size criterion implied in the intuitive notion of containers, a container A can vary its size over time as to squeeze itself into another container B so that it become a member of B, and similarly B can vary its size as to squeeze itself into container A at some *other* moment of time. So we can imagine a sort of platonic time-space complex in which all these containers exist. So: -------------------------------------------------------- A Set is a content specific container. -------------------------------------------------------- COMPARISON BETWEEN THE TWO LINES OF THOUGHT. An interesting point is that the container interpretation is temporally explained, while the collection status interpretation is static, i.e. includes all sets are formed at the same time, so there is no time concept involved. The collection status line of though is linguistic, it is conceptual and though more difficult to grasp, but yet seem to me to be more convincing. On the other hand the container line of thought is visual and temporal, it is dynamic, it is objective rather than conceptual, it imagines sets as objects in imaginary time-space complex, rather than considering them as simpler concepts as the collection status line of thinking does, it is less convincing, but yet easier to grasp. The collection status line of thought seems to be explanatory, while the container line of thought seems to be illustrative. This subject is the cause of wonders, it is a wonderland no much different from that of Alice in Wonderland! Zuhair
From: zuhair on 10 Feb 2010 23:59 On Feb 10, 11:31 pm, zuhair <zaljo...(a)gmail.com> wrote: > What are sets? I am interested to read about earlier attempts at this particular subject. Can anyone mention some references in which this particular subject is dealt with in a thorough,comprehensive, and detailed manner. Thanks in advance. Zuhair
From: zuhair on 11 Feb 2010 00:02 On Feb 10, 11:31 pm, zuhair <zaljo...(a)gmail.com> wrote: > What are sets? > > Can we have a simple intuitive understanding of these objects > called as sets by mathematicians, like those objects in Z > set theory or Aczel's, etc.. > I am interested to read about earlier attempts at this particular subject. Can anyone mention some references in which this particular subject is dealt with in a thorough,comprehensive, and detailed manner; and especially if there such a reference available on-line. Thanks in advance. Zuhair
From: zuhair on 12 Feb 2010 02:25 On Feb 10, 11:31 pm, zuhair <zaljo...(a)gmail.com> wrote: > What are sets? > > Can we have a simple intuitive understanding of these objects > called as sets by mathematicians, like those objects in Z > set theory or Aczel's, etc.. > > Here is a simple trial on my side to understand > those rather vague objects intuitively. > > First lets intuitively interpret sets as *collections*. > > THE COLLECTION INTUITIVE LINE OF THOUGHT:- > > Collections arises most clearly when we come across > *multiple* objects. *collections* as a concept arises > when we deal with multiple objects as ONE entity, > and by then we describe the relationship between > those objects and this one entity as > those objects being members of that entity, > from that line of imagination we can then > extend the concept of collections to cover > single objects, but can we explain the empty set? > > The above was the context in which the term *collection* > is mostly used in common language. > > Sets are often thought as collections, the term collection > is an informal term, it is not well defined. > > Let's try to go to our world and take simple examples of what we > may call as collections, let's examine how they are build and give > relevant terminology. > > "To collect an object" , frequently refers to, > "putting an object into a container" > > Matters start usually by an attempt to collect objects, or what we > may refer to as "collective attempt". Now the collective attempt > is a term that describes the process of all trials to collect objects > under certain given conditions and circumstances, so according to > the understanding here it would be a term that describes the > process of all trials to put objects into a container as specified by > conditions of the attempt. > > Now collective attempts vary according to the condition required for > collecting objects, for example a collective attempt to collect > money, is different from that one that aims at collecting a special > kind of sea food. > > Not only that, collective attempts also vary in many other details, > however two collective attempts might result in the same objects > being collected. > > What we care about here is the "end result" of the collective > attempt, and here we don't care about the various details that lead > to these end results. > > Now we have two main end results of collective attempts. > > A failed collective attempt > > A successful collective attempt > > The failed collective attempt, is a collective attempt that didn't > result in any object being collected, i.e. no object is put in the > container, so the container into which objects are to be put, > remained empty. > > A successful collective attempt is one that succeeded in putting at > least one object into the container. > > Now we come to define what we mean by the term "collection". > > ---------------------------------------------------------------------- > A collection is the whole of all collected objects. > ---------------------------------------------------------------------- > > If there is no collected objects, then there is no whole of them, > then there is no collection. > > A collection results from a successful collective attempt only! > > A failed collective attempt results in "no collection" inside the > container. > > So there no empty collection! > > If the collective attempt resulted in only One object being collected, > then the whole of all collected objects would be > that object itself! So a collection of one object > is that object. > > If we want to make a formal system about *collections*, then > it would be something like the following: > > let's denote the membership relation by the symbol "e", > then we'll have the following formal system in FOL > with identity and e. > > (1)Two collections having the same elements are identical. > (2)The empty collection do not exist. > (3)The singleton collection is identical to its element. > (4)Every element of a collection is singleton. > (5)For any formula Phi in which x is not free, then > all closures of > > For all y ( Phi(y) -> y is a singleton) and Exist y Phi(y) > -> > Exist x for all y ( y e x iff Phi(y) ) > > are axioms. Criticism: The above system is not necessarily the case always! Actually Only the first two axioms are intuitively appealing, the other three not so evident. The third axiom appears intuitive if we deal with singleton collections of physical objects, but if we allow collections themselves to be members of collections then we can intuitively imagine a situation in which this axiom fail, take a collection of two elements {x,y}, now take the singleton of this collection {{x,y}}, then it is clear that the later is no identical to it's element, since its element is not a singleton. This would lead to a tautology of A singleton collection is identical to its member if and only if it contain no member other than itself. which is reduced to A singleton collection is identical to its member if and only if it is identical to its member. A clear tautology. This tautology mean that there is no necessary relationship between a collection being singleton and it being identical to its member, a collection might be singleton and be identical to its member, or it can be otherwise. since a collection is an object, then there is no intuitive background for restricting it from being collected, so this mean there is no sufficient intuitive grounds for axiom 3. The above would actually lead to the intuitive possibility of separation between collection and its members, so a collection is an entity that can be separate from its members, so membership relation is not a special kind of part-hood relation, which makes the collection concept depart from Mereology. Now can one stretch the collection concept such as to say that there exist an empty collection, the rational beyond that is that seeing that axiom 3 can be violated intuitively and a collection can be a separate entity from its members, then we can go further to postulate that the existence of a collection do not depend on the existence of its members, and thus laying the grounds for accepting an empty collection. However the above extrapolation to justify the existence of an empty set is not intuitive at all especially if we are defining a collection as being a whole of objects, still as anyone can see the intuitive statement of *if there are no objects then there is no whole of them* seems to preclude the above rather technical extrapolation of the collection concept to include the empty case. so if we drop the last three axioms of the system above, the gap between a collection status and a collection would greatly narrow as to enable us to postulate that a successful collection status is a collection, and a collection is a successful collection status. So we can define: A collection is a successful collection status. However the failed collection status still represent a challenge to be thought of as a collection. Still the original definition of a set as a collection status holds, but the difference here is that non empty sets are collections and collections are non empty sets. However the difference is very subtle now, that for the sake of technicality it seems justifiable to extend the concept of collections to involve the failed collection status itself. However to be more precise, the term "collection status" is the essential one here, so Sets are collection statuses. SELECTION: If we are pursuing to explain *sets* intuitively, then it appears to me that, the term *selection* is more appropriate then *put in a container* to explain what the act "to collect* refers to, since in set theory recognition of an object to fulfill a specified condition results always in it being enrolled in the set, so this is equivalent to saying that if any object meets the requirements for being selected then it is successfully been put into the container, of course in the physical word this is not necessarily the case, but working with sets, it is always the case, so it seems that we can replace the word "collect" by the word "select" Selection refers to "picking an object from among many objects" Accordingly one might define a collection as A collection is the whole of all selected objects. Also the collective attempt can be rewritten as: *collective attempt* is a term that describes the process of all trials to select objects under certain given conditions and circumstances. We can have three main types of selections: (1) Conditional selection: objects are selected according to fulfillment of a specified clear well defined condition, like in saying if x is a human then x is selected and so on... (2) Random selection: objects are arbitrarily selected without any condition specified. (3) Ambiguous selection: were objects are selected according to some hidden condition that is not describable clearly, i.e. not well defined condition, like for example the set of all people that Samantha loves. here love is a subjective feeling it is not defined in a clear cut manner. Most selections that set theory matters with is of the first two kinds, which is Constructive selection and Choice selection respectively. Regards. Zuhair > > Which is clearly quite different from the formal systems > of Sets like Zermelo's set theories and others. > > So sets are not collections in the above sense. > > Now I come to explain what I mean by a "collection status" > > A collection status is the end result of the collective attempt. > > More specifically a *collection status* is a term that describes the > end result of collective attempt in a differential manner, so if the > collection status of a failed collective attempt is named as "K" > and the collection status of a collective attempt that succeeded in > collecting the object B is to be named as "H", then K must not be > identical to H. So two collection statuses are identical if and only > if the end result of the collective attempts is the same, > i.e. either it is *failure* for both of the collective attempts, > or both are successful resulting in the same collection. > Accordingly any two collection statuses would be different > if and only if one failed while the other succeeded > or both succeeded but resulting in two different collections. > > Now is the collection status an object? > > The collection status looks as a situation, however this can be > looked as an object although definitely of a nature that is quite > different from observed objects in our world. > > So we can understand sets as collection statuses. > > So: > > A set is a collection status. > > So sets are not collections, sets are collection statuses! > > Now we come to explain what we mean by the "membership" > relation: > > A is said to be a member of X, if and only if, > X is the collection status that describes the end result of > a collective attempt having A successfully collected (put into > the container). > > The above line of intuitive thinking as to the nature of sets > and their members would allow the existence of an empty set, > and of sets being members of sets, and of building up higher > hierarchy of sets.it serves a strong intuitive basis for simple type > theory, and well founded set theories. > > However the above line of thinking cannot easily > explain non well founded sets, the reason is > our intuitive habit of imagining the collection status > to be formed *after* the end of the collective attempt. > Accordingly a set cannot be a member of itself, > since this would imply that it started to be formed after > it was already formed, which is ridiculous. > > Another source of difficulty is our habit of imaging sets > of physical objects, like sets of apples, of humans, etc.., > now the objects in these sets are definitely different > in nature from "collection statuses", those objects > are "observed entities", while "collection statuses" > are descriptive entities, the later are concepts rather > than physical objects, and if we are to regard > collection statuses as objects, then they are > definitely of a very different nature than those > observed entities, this lead to the impression > that sets are of different kind from their members, > and so a set cannot be in itself or in any of the > members of its transitive closure, since this entail > that it would have a nature that is different from its > nature, which is ridiculous. > > However if we imagine that we are working with > pure sets, i.e. sets in which every member of their > transitive closures are sets, then all these sets > are of the same nature, all of them are "collection statuses", > so the difficulty that arise with sets of physical objects > is not longer a problem. > > Also there is nothing to imply the existence of a time factor, > A collection status is a *description* of the end result > of the collective attempt, but one can imagine that, the collection > status and the objects collected all are timeless objects, since > both the collected objects and the collection status all are > descriptions then there is no real reason for regarding the > existence of any time factor in the sense that these descriptions > are *formed* in time, and that there is a time lag between them, all > these sets are concepts, they are objects yes, but they are > conceptual objects, and not a visual-temporal objects like that of > the real world. Seeing that pure sets are of the same nature, and > that there is no time factor involved, then there is nothing against > the existence of non well founded sets. > > So the above was the collection status line of thought. > > So the main result is that: > > ---------------------------------------- > A set is a collection status. > ---------------------------------------- > > Now I'll introduce another quite different approach than the above, > which is the container line of thought. > > THE CONTAINER INTUITIVE LINE OF THOUGHT:- > > It appears that the easier way to explain sets including non > well founded sets intuitively is to think of sets > as *containers*. > > Sets can be defined as *content specific containers*, > > This can be defined formally as: > > x is a set iff for all y ( for all z ( z e x iff z e y ) -> x=y ) > > so if x is a container such that there do not exist > any other container having exactly the same contents as x, > then x is a set, and vise versa, i.e. if x is a set then > x is a container such that there do not exist any other > container having exactly the same contents as x. > > While on the other hand distinct containers having > the same contents are not sets, they can be called > as *para-sets*, but in any case they are not sets. > > If we imagine sets according to the above intuitive line, > then it is not difficult at all to imagine a container that > is in itself (a spiral), or in one of its members, > or in one of the members of its transitive closure, > also it is not difficult to imagine a never ending chain of > containers one in each other, all these are possible to imagine. > > We can even define containment in a visual-temporal way, > so A contained in B, which is A e B , if and only if, > all parts of A are surrounded by parts of B at > some moment of time. > > So if we imagine A as infinite spiral, > then every part of A would be surrounded > by a part of A, actually all moments of time, > so A would be in itself. > > We can also follow another line of imagination, > though more dynamic, in which we > can postulate a time period on membership ... > > read more »
From: Michael Stemper on 12 Feb 2010 13:17
In article <dc6c5043-8e47-48a5-83e7-d3642f888adf(a)k41g2000yqm.googlegroups.com>, zuhair <zaljohar(a)gmail.com> writes: >On Feb 10, 11:31=A0pm, zuhair <zaljo...(a)gmail.com> wrote: >> What are sets? > >I am interested to read about earlier attempts at this particular >subject. > >Can anyone mention some references in which this particular subject is >dealt with > >in a thorough,comprehensive, and detailed manner. It doesn't get much more thorough, comprehensive, or detailed than in _Axiomatic Set Theory_ by Patrick Suppes. Chapter 1 and much of Chapter 2 are available on-line at: <http://books.google.com/books?id=sxr4LrgJGeAC&dq=Axiomatic+Set+theory+Patrick+Suppes&printsec=frontcover&source=bn&hl=en&ei=SZp1S770MYm0tgfihoyqCg&sa=X&oi=book_result&ct=result&resnum=4&ved=0CBcQ6AEwAw#v=onepage&q=&f=false> If you can get through this much, then buy the book, which is available as a Dover reprint for $12.95 (new). -- Michael F. Stemper #include <Standard_Disclaimer> Nostalgia just ain't what it used to be. |