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From: John Jones on 27 Dec 2009 00:40 Virgil wrote: > In article <hh3m28$8ii$1(a)news.eternal-september.org>, > John Jones <jonescardiff(a)btinternet.com> wrote: > >> zuhair wrote: >>> Hi all, >>> >>> One of the facts of set theory is that the primitive relation >>> "e" is totally vague relation, there is no way of understanding this >>> relation at all, even at informal level! Which is actually a great >>> drawback, since in set theory we are building hierarchy of objects >>> using this relation, a relation that nobody understand what it stands >>> for? >>> >>> Here I will present a simple trial to understand this relation "e" >>> and to give it some informal intuitive background. >>> >>> First I shall speak about the concept of set as used in ordinary >>> language. >>> >>> When we say we have a set of objects x,y,z , then we usually mean that >>> there is an entity that represent the *Whole* of >>> x,y,z, so although x,y,z are distinct objects, but yet >>> the *whole* that is constituted from these objects is considered the >>> set of these objects, or the *collection* of these objects. >>> >>> However this *Whole* of x,y,z is in a sense these objects themselves. >> So far so good, but that last statement is incorrect. There's no >> relationship between parts and their whole. >> >>> So a set is a form of describing multiplicity by unity, that unity is >>> taken to be the totality of the Whole of these multiple objects. >>> >>> However if we collect ONE object, then it is pretty intuitive to say, >>> that the set of all objects collected IS that object. >> The set "the set of all objects collected" doesn't look like a set to >> me. It begs the question of what a set is, and if its a collection then >> it looks tautologous. It just says "set"? >> >>> Also it is pretty intuitive to say that if there are no objects >>> collected, then there is NO set of objects collected, in other words >>> there do not exist an empty collection. >>> >>> Of course the traditional membership relation "e" doesn't capture all >>> these pretty intuitive concepts of the customary use of the word >>> membership and of the word "set" or the word "collection". >>> >>> To try to put all of the above informal account into rigorous >>> mathematical context, we can define a theory in FOL with identity >>> and the primitive binary relation symbol e' , which stands for >>> what I call "intuitive membership", or in reality "membership", since >>> epsilon doesn't capture the intuitive concept of membership. >>> >>> So we can have the following axioms that characterize this binary >>> relation e'. >> I don't think you will find the resources you need from what is >> currently available to you. I would look at the fringes. > > A necessary and sufficient realization of the idea of a "set" is the > ability to distinguish between the objects one wishes to include in the > set from tho objects one wishes to exclude. > > I.e., if one can deal successfully with the issue of membership, then > one has grasped the idea of "set". But neither membership nor distinguishment identify a set.
From: Virgil on 27 Dec 2009 01:39 In article <hh6rvu$urn$2(a)news.eternal-september.org>, John Jones <jonescardiff(a)btinternet.com> wrote: > Virgil wrote: > > In article <hh3m28$8ii$1(a)news.eternal-september.org>, > > John Jones <jonescardiff(a)btinternet.com> wrote: > > > >> zuhair wrote: > >>> Hi all, > >>> > >>> One of the facts of set theory is that the primitive relation > >>> "e" is totally vague relation, there is no way of understanding this > >>> relation at all, even at informal level! Which is actually a great > >>> drawback, since in set theory we are building hierarchy of objects > >>> using this relation, a relation that nobody understand what it stands > >>> for? > >>> > >>> Here I will present a simple trial to understand this relation "e" > >>> and to give it some informal intuitive background. > >>> > >>> First I shall speak about the concept of set as used in ordinary > >>> language. > >>> > >>> When we say we have a set of objects x,y,z , then we usually mean that > >>> there is an entity that represent the *Whole* of > >>> x,y,z, so although x,y,z are distinct objects, but yet > >>> the *whole* that is constituted from these objects is considered the > >>> set of these objects, or the *collection* of these objects. > >>> > >>> However this *Whole* of x,y,z is in a sense these objects themselves. > >> So far so good, but that last statement is incorrect. There's no > >> relationship between parts and their whole. > >> > >>> So a set is a form of describing multiplicity by unity, that unity is > >>> taken to be the totality of the Whole of these multiple objects. > >>> > >>> However if we collect ONE object, then it is pretty intuitive to say, > >>> that the set of all objects collected IS that object. > >> The set "the set of all objects collected" doesn't look like a set to > >> me. It begs the question of what a set is, and if its a collection then > >> it looks tautologous. It just says "set"? > >> > >>> Also it is pretty intuitive to say that if there are no objects > >>> collected, then there is NO set of objects collected, in other words > >>> there do not exist an empty collection. > >>> > >>> Of course the traditional membership relation "e" doesn't capture all > >>> these pretty intuitive concepts of the customary use of the word > >>> membership and of the word "set" or the word "collection". > >>> > >>> To try to put all of the above informal account into rigorous > >>> mathematical context, we can define a theory in FOL with identity > >>> and the primitive binary relation symbol e' , which stands for > >>> what I call "intuitive membership", or in reality "membership", since > >>> epsilon doesn't capture the intuitive concept of membership. > >>> > >>> So we can have the following axioms that characterize this binary > >>> relation e'. > >> I don't think you will find the resources you need from what is > >> currently available to you. I would look at the fringes. > > > > A necessary and sufficient realization of the idea of a "set" is the > > ability to distinguish between the objects one wishes to include in the > > set from tho objects one wishes to exclude. > > > > I.e., if one can deal successfully with the issue of membership, then > > one has grasped the idea of "set". > > But neither membership nor distinguishment identify a set. But they separate what "belongs" to the set from what doesn't. You might think of a set as a boundary, like a simple closed curve in the plane, having an inside and an outside, but unlike such curves, not having things as a part of that boundary.
From: zuhair on 27 Dec 2009 09:40 On Dec 26, 1:38 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Dec 24, 7:41 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > Hi all, > > > One of the facts of set theory is that the primitive relation > > "e" is totally vague relation, there is no way of understanding this > > relation at all, even at informal level! Which is actually a great > > drawback, since in set theory we are building hierarchy of objects > > using this relation, a relation that nobody understand what it stands > > for? > > This is completely untrue. A simple, "intuitive" understanding > of of the is-an-element-of relation is trivial, even for middle > school children; they are taught about sets and it is quite > easy stuff. At the formal level, we have the axioms. > > I can see no basis whatsoever for these claims. Well why you don't just tell me what is this what you call "intuitive" meaning of the membership relation epsilon is? Let me give you some issues: If you say that a set is an aggregate of it's members (Review Bertrand Russell in Introduction to mathematical philosophy) which is by the way the trivial meaning of a "set" as used in customary everyday language, then you will end up with the difficulty of having a singleton that is not its member, and also you will have the catastrophic difficulty of the empty set. These issues were discussed almost one century ago, and nobody came to answer this issue. Now if you say that a set is a "container" and the relationship epsilon stands for "being contained in" or " being inside" or something like that, you'll have the difficulty of why should containers be identical if there contents are identical? that doesn't make any sense at intuitive level, intuitively speaking different containers of the same size can contain the same elements, there is no reason whatsoever to believe that two containers having the same contents cannot be different, which is what axiom of Extensionality says. On the other hand, having an *empty* set, cannot be explained intuitively other than by saying that what is called as "set" is in reality some sort of a "container" since only "containers" can be empty! Same thing is applicable with singleton sets that are not identical to their members, also these can be though of as "containers" containing one member. There is another difficulty with the "container" intuitive background of sets, and that is:_ we have strong theories allowing 'sets' to be in themselves like Aczel's set theory, and this is somewhat difficult to explain by the container theory, since this entail the existence of a container that contain itself, although this is not an absolute difficulty against the container concept, but yet it demands intuitive justification of containers containing themselves and the example of a snake that swallows itself is an example of how a container might be in itself, but yet it is not so clear, and it needs further treatment. Here in this account I tried to capture both intuitive backgrounds of membership relations and give them separate symbols, the first background which is that of common language use of the word member and set, which I symbolized as e' which is what I called "intuitive membership" or "trivial membership", and the axioms I stipulated do really capture this intuitive membership, so we can speak of "sets" as used in customary language more intuitively with that axiomatic system, and the collections that use this membership are what I call "true collections". The second background is the relation epsilon "e" that is used by set theoriests, and I gave it the meaning of "being a member (by relation e') of the largest possible collection of discriminable containers" However we might not have such "largest collection of discriminable containers" so we might define sets according to the collection of discriminable containers they are trivial members of: So we might say for example x is a V_set if and only if [V (which might be stipulated as a primitive constant) is a true collection of discriminable containers AND x e' V] So we might not be able to define the word "set" in an absolute manner, perhaps the word "set" must be defined relatively, i.e. relative to the collection of discriminable containers it is a trivial member of. However if we have for example, a collection V that is maximal, i.e. for any collection Vi that is a collection of discriminable containers then Vi is smaller than V. Were smaller than can mean "a true sub-collection of " Or it may mean "subnumerous to" ( for that we need to stipulate cardinality as a primitive one place function symbol, and axiomatize it to be equal only for equinumerous true collections) This is a long subject that I myself didn't finish it. However If we succeed with the maximal notion, then we can define a set as a trivial member of that maximal collection of discriminable containers. Any by then we can say we do HAVE a definition of "set". Would it make a difference, I don't know. For instance this approach can define a proper class of *ALL* ordinals something that all current approaches to membership cannot do, doesn't it? On the other side, it might not be useful as you said. However as a last word that I want to say here: Even this approach is not completely a pure intuitive explanation of what sets are, it is a mixed formal-intuitive explanation. But it affords a sort of an explanation. Zuhair > > > However if we collect ONE object, then it is pretty intuitive to say, > > that the set of all objects collected IS that object. > > It is a matter of choice. It we think of sets as just the > collections of elements, maybe. If we think of sets as > containers, then no. It is not difficult to distinguish between > a social club with one member and the person himself. > Even intuitively. > > > Also it is pretty intuitive to say that if there are no objects > > collected, then there is NO set of objects collected, in other words > > there do not exist an empty collection. > > Whether it is intuitive or not is debatable and unlikely to > be of much importance except from a pedagogical view. > I subscribe to the oft-quoted view that the only intuitive > concept is to suck on the nipple, and anything else is > learned. > > > Of course the traditional membership relation "e" doesn't capture all > > these pretty intuitive concepts of the customary use of the word > > membership and of the word "set" or the word "collection". > > > To try to put all of the above informal account into rigorous > > mathematical context, we can define a theory in FOL with identity > > and the primitive binary relation symbol e' , which stands for > > what I call "intuitive membership", or in reality "membership", since > > epsilon doesn't capture the intuitive concept of membership. > > Trying to build systems by extrapolating from natural language > has been attempted many times, and has never yielded anything > fruitful. It is much more successful to consider things like > expressive power, simplicity, elegance, etc. I think your > choice of design criteria is fundamentally poor; even if you > succeed in meeting your criteria, you still won't have anything > useful. > > Marshall
From: Marshall on 27 Dec 2009 12:51 On Dec 27, 6:40 am, zuhair <zaljo...(a)gmail.com> wrote: > On Dec 26, 1:38 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > On Dec 24, 7:41 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > One of the facts of set theory is that the primitive relation > > > "e" is totally vague relation, there is no way of understanding this > > > relation at all, even at informal level! Which is actually a great > > > drawback, since in set theory we are building hierarchy of objects > > > using this relation, a relation that nobody understand what it stands > > > for? > > > This is completely untrue. A simple, "intuitive" understanding > > of of the is-an-element-of relation is trivial, even for middle > > school children; they are taught about sets and it is quite > > easy stuff. At the formal level, we have the axioms. > > > I can see no basis whatsoever for these claims. > > Well why you don't just tell me what is this what you call "intuitive" > meaning of the membership relation epsilon is? It means "is an element of." At the middle school level, we can easily have a conception of sets as collections of arbitrary elements, and we can define operations on sets like union and intersection. Union and intersection are no more difficult than logical "or" and "and." But in any event, I reject the idea that studying what one currently does or does not find intuitive is worth spending any time on. Rather, it is productive to train one's intuition to better match theory. It is theory that represents the best understanding of mathematics, not intuition; if they conflict, then intuition must improve. Of course, you may object that there are times when new theories are born. However these times are not as a rule the result of considering trivia such as whether the existence empty set violates some unspecified intuitive sense. Note that it is not your statements merely that I object to, but your very methodology. You are thinking about these issues in the wrong way. The considerations that are likely to be useful in your pursuit are things like simplicity, expressive power, utility. Sometimes I hear the term "power-to-weight ratio." Think about what you want to accomplish with the theory, about what you want to be able to do. Is it easier with your new approach or with the established ways? Unless you can give some practical benefits to your approach, you haven't found anything worth attention. "I like this way better" is unlikely to be practical enough. > Let me give you some issues: > > If you say that a set is an aggregate of it's members (Review > Bertrand Russell in Introduction to mathematical philosophy) > which is by the way the trivial meaning of a "set" as used > in customary everyday language, then you will end up with > the difficulty of having a singleton that is not its member, and > also you will have the catastrophic difficulty of the empty set. > These issues were discussed almost one century ago, > and nobody came to answer this issue. I reject the idea that these are issues. I reject the relevance of everyday language. > Now if you say that a set is a "container" and the relationship > epsilon stands for "being contained in" or " being inside" > or something like that, you'll have the difficulty of why > should containers be identical if there contents are identical? I see no difficulty. > that doesn't make any sense at intuitive level, intuitively > speaking different containers of the same size can > contain the same elements, there is no reason whatsoever > to believe that two containers having the same contents cannot be > different, which is what axiom of Extensionality says. These "difficulties" are actually virtues, and whether you want to work with a system that has these properties or not, until you see why they are virtues, you are not going to get anywhere. Consider a system where, in addition to its elements, every instance of a set also has identity. Then we go from having one kind of equality to two kinds: equality of members and equality of identity. The system is now vastly more complicated. Is there some corresponding vast improvement in some other dimension to make up for this increase in complexity? No there is not. What math does, what logic does, is to abstract. To abstract is fundamentally to leave things out; this is inescapable. If you want to keep everything in, use the actual universe itself as your system. This is the smallest system that will capture *all* the behavior of the world around you. If you want to represent three oranges, then you must use three oranges to do that, in fact you must specify which specific three oranges you mean, because you apparently care about identity. If you want to know how many oranges you will have when you add another orange, you must first specify which particular orange you mean. The result, then, is those exact oranges you specified, and we cannot say anything less than that. Generalization is impossible, because generalization necessitates leaving things out. On the other hand, if you just want to know what 3+1 is, then you are necessarily leaving out a lot of details about the oranges. You are abstracting away those details. This is a virtue; it means your results will be widely applicable. > On the other hand, having an *empty* set, cannot be explained > intuitively other than by saying that what is called as "set" is in > reality some sort of a "container" since only "containers" can be > empty! Unless you have some trouble with the idea that 0 is a number, I see no reason for you to have any difficulty with the idea that the empty set is a set. > Same thing is applicable with singleton sets that are not identical to > their members, also these can be though of as "containers" containing > one member. > > There is another difficulty with the "container" intuitive background > of sets, and that is:_ we have strong theories allowing 'sets' to be > in themselves like Aczel's set theory, and this is somewhat difficult > to explain by the container theory, since this entail the existence of > a container that contain itself, although this is not an absolute > difficulty against the container concept, but yet it demands intuitive > justification of containers containing themselves and the example of a > snake that swallows itself is an example of how a container might be > in itself, but yet it is not so clear, and it needs further treatment. I reject the suggestion that anything "demands intuitive justification" and say again that your thinking in these terms means that your thinking is fundamentally broken. (But entirely fixable.) Again, you are thinking in excessively concrete terms. If the only way you are able to think about containers is by analogy with a grocery bag, then you are in serious conceptual trouble. Writing a computer program in which a data structure contains a reference to itself is a trivial exercise. So why should this present any difficulty to set theory? > [...] > On the other side, it might not be useful as you said. I propose that you *start* with utility or simplicity or some such virtue and proceed from there. Marshall
From: Nam Nguyen on 27 Dec 2009 17:02
zuhair wrote: > Hi all, > > One of the facts of set theory is that the primitive relation > "e" is totally vague relation, there is no way of understanding this > relation at all, even at informal level! Which is actually a great > drawback, since in set theory we are building hierarchy of objects > using this relation, a relation that nobody understand what it stands > for? > > Here I will present a simple trial to understand this relation "e" > and to give it some informal intuitive background. > > First I shall speak about the concept of set as used in ordinary > language. I think it's a bad idea to assert that there's a unique "set" binary relation you've referred to as 'this relation "e"' that doesn't any intrinsic vagueness in it. So long as we talk about a kind of set formal systems that could be used to encode the naturals (which most common set theories - as well as your alluded set theories - are) then there's always vagueness in these set concepts. For instance, is there any clarity of the truth of "There are infinite counter examples of GC" in these set systems? One certainly could make the set concept more implicated (e.g. by introducing multiple binary epsilon relations) but I think one should abandon the idea that one would have "better" explanation (as you've tried in this thread) of what set concept is than the existing set formal systems, given the constraint/limitation mentioned above. > > When we say we have a set of objects x,y,z , then we usually mean that > there is an entity that represent the *Whole* of > x,y,z, so although x,y,z are distinct objects, but yet > the *whole* that is constituted from these objects is considered the > set of these objects, or the *collection* of these objects. > > However this *Whole* of x,y,z is in a sense these objects themselves. > > So a set is a form of describing multiplicity by unity, that unity is > taken to be the totality of the Whole of these multiple objects. > > However if we collect ONE object, then it is pretty intuitive to say, > that the set of all objects collected IS that object. > > Also it is pretty intuitive to say that if there are no objects > collected, then there is NO set of objects collected, in other words > there do not exist an empty collection. > > Of course the traditional membership relation "e" doesn't capture all > these pretty intuitive concepts of the customary use of the word > membership and of the word "set" or the word "collection". > > To try to put all of the above informal account into rigorous > mathematical context, we can define a theory in FOL with identity > and the primitive binary relation symbol e' , which stands for > what I call "intuitive membership", or in reality "membership", since > epsilon doesn't capture the intuitive concept of membership. > > So we can have the following axioms that characterize this binary > relation e'. > > (1) Extensionality: > For all z ( z e' x <-> z e' y ) -> x=y > > Define(singleton): > x is singleton <-> Exist z for all y ( y e' x <-> y=z ) > > (2) Membership: x e' y -> x is singleton > > (3) Singletons: x is singleton -> x e' x > > theorem: x is singleton <-> x e' x > Proof: we have x e' x -> x is singleton (2) > and we have x is singleton -> x e' x (3) > thus x is singleton <-> x e' x > QED > > (4) Non Emptiness: For all x Exist y ( y e' x) > > (5) Comprehension: if Phi(y) is a formula in which at least y is free, > and in which x is not free, then all closures of > > Exist y ( y is singleton Phi(y) ) -> > Exist x for all y ( y e' x <-> ( y is singleton Phi(y) ) ) > > are axioms. > > Define[y|Phi]: > Exist y ( y is singleton Phi(y) ) -> > x=[y|Phi] <-> for all y ( y e' x <-> ( y is singleton Phi(y) ) ) > > Now we come to explain what "sets" in set theory mean: > > The approach here is that objects in a set theory like Zermelo's set > theory, are actually nothing but "containers", that is their reality. > > However these containers in Z, have the property of being identical if > they have the same contents, which is not a property that is inherent > in the concept of containers, since containers of equal size can have > exactly the same contents in them but yet can be different from each > other, however this is not allowed in Z, so how come I say that these > objects are containers. > > To understand that issue, lets assume that sets are containers, and > lets assume the relation epsilon in the language of Z refers > informally to "contained in" so x e y means informally > x is contained in y. > > Of course "e" here is a primitive concept also, but in this approach > it has an informal background as I illustrated above, it refers to > "containment". > > Now lets have the following axiom: > > (6) Containment: x e y -> ( x is singleton & y is singleton ) > > were singleton is defined above ( note: the definition of singleton > uses the relation e' so it is not the same as singletons present in > customary set theories which uses the relation e). > > e can be read as " is a content of" > > so x e y is to be read as: x is a content of y. > > Now we define any collection of containers in which no two distinct > containers have the same contents as a "discriminable collection" > > Now the union of two discriminable collections is not necessarily > a discriminable collection. > > So we need to define discriminable union as the union of any two > discriminable collections that is itself discriminable. > > Now lets define V as the largest discriminable union of > discriminable collections (this notion needs further treatment as to > what "largest" would mean, but for now it means that every > discriminable union of discriminable collections would be strictly > subnumerous to V) > > Now a 'set' might be defined as a member of V. > > So V here would stand as the model of that set theory. > > Then other axioms of Z, or ZF can be added ( were the variables in > these axioms are members of V ). > > This way of understanding sets, although it is not a complete workup > yet, but it does provide us with an insight to what are sets, what are > those objects in ZFC set theory. > > They are Containers discriminable by their contents, nothing more, > nothing less. > > The relation e' can be used here to build proper classes also. > > So it is a theory of this theory that: > > For all x,z,u,w ((u=[y|yex] & w=[y|yez] & u=w) ->x=z) > > Zuhair |