The set of All sets
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From: zuhair on 29 Nov 2005 13:11 Jonathan Hoyle wrote: > >> And what is that framwork. > > Typically ZFC is the default, unless stated otherwise. You can move to > NBG, which is proven to be equi-consistent. But point is: unless you > move to a more exotic logical framework, there is no such thing as the > set of all sets. > > >> There is something missing. I don't like the reply you've made. > >> It is a kind of a fixed answer to defend the current practice. > > I am not defending a "practice" as much as I am explaining the logical > foundations. I'm sorry you do not like the reply. > > >> you should define collection in a clear manner. > > "Collection" is a meta-theory term. Some collections are "sets" within > a framework, others are not. When a collection is not a set within a > framework, it is not addressable within that framework. For example, > your "collection of all sets" is clearly not a set in ZFC, as it > violates one of its axioms, namely the Axiom of Foundation (AF). In > logical frameworks which do not include AF, it could be a set, but you > would have to tell me what framework you are using. > > If the meta-theoretical terms bother you, fine. Then we can tersely > sum it up this way: assuming ZFC, there is no such thing as the set of > all sets. Case closed. > > Jonathan Hoyle > Eastman Kodak What you are saying is simple and easy to understand. According to some logical framework the collection of all sets is a set and according to another logical framework it is not a set. Is there a standard of preferece between these frameworks regarding that case, I mean specifically regarding weather the collection of all sets is a set or not. I think the word "collection" should have a seperate definition from any framework, so that we can manage to form a preference standard between framework that can tell us weather the collection of all sets is a set or not. collection should be something more primitive than a set, in such a way that every set is a collection but not the converse. And we should say for example that a set is a collection having the so and so criteria. and any collection that do not have the so and so criteria is not a set. And that standard should be the same for all logical frameworks. The existance of different logical frameworks indicate a sort of unsolvable question(s). Zuhair
From: Jonathan Hoyle on 29 Nov 2005 15:44 >> What you are saying is simple and easy to understand. According >> to some logical framework the collection of all sets is a set and >> according to another logical framework it is not a set. >> >> Is there a standard of preferece between these frameworks >> regarding that case, I mean specifically regarding weather the >> collection of all sets is a set or not. Typically, ZFC is assumed as the logical framework, unless you state otherwise. It's like speaking about geometry, and it is assumed to be Euclidean geometry unless you state otherwise. Both geometries are equi-consistent, but many of the theorems will have different results. For example, someone can say "the sum of the angles of a triangle is 180 degrees" and not need to specify that this is true in Euclidean geometry but false in Non-Euclidean, since it is assumed. NBG Set Theory is a favorite of ZFC proponents, in that virtually all of the theorems of ZFC hold true in NBG as well. NBG however introduces extensions, such as the concept of proper classes. NBG terminology has become so common that many people use it implicitly as well. For example, many mathematicians prefer to say, "the class of all ordinals is not a set" (NBG-correctness) as opposed to "the set of all ordinals does not exist" (ZFC-correctness). Using more exotic (non-Well Founded) Set Theories will yield very different results. For instance, I do not know how cardinality is defined on non-WF sets (if it is at all), but it would certainly have to be different than what we have in ZFC. >> I think the word "collection" should have a seperate definition >> from any framework, so that we can manage to form a >> preference standard between framework that can tell us >> weather the collection of all sets is a set or not. The term "collection" is typically used from a "naive" meta-Set Theory perspective. Since it exists outside the framework, it cannot be defined within it. To create a meta-definition for it, a meta-logical framework for Set Theories would have to be defined first. However, the term is also used to refer to inconsistent predicated groups, such as "the Russell collection", which are things which cannot exist at all, and thus no consistent definition would apply. >> collection should be something more primitive than a set, in such >> a way that every set is a collection but not the converse. Conceptually, that is what we mean it to be, but once we give it a firm definition, it will likely have to exclude certain collections (such as the inconsistent ones, like Russell's). For "collection" to exist outside of any axiomatic logical framework, a firm definition may be elusive. >> And we should say for example that a set is a collection having >> the so and so criteria. and any collection that do not have the so >> and so criteria is not a set. And that standard should be the >> same for all logical frameworks. This is certainly a laudable goal, but remember that once we are outside a given logical framework, we have no basis to "prove" if a collection is a set. This is the main problem. The typical approach to work the other way around: assume a given collection is a set within a logical framework, and see if you can derive a contradiction. If so, then the collection is not a set in that framework. I agree that this is a bit unsatisfying compared to your "A set in Theory X is a collection with the following attributes...", but outside a formal system, you cannot formalize such a proof. I am curious if this has been studied by metalogicians and metamathematicians. Most of the books I have seen on Metalogic and Metamathematics are more a study of Proof Theory than they are a study of comparative Set Theory. >> The existance of different logical frameworks indicate a >> sort of unsolvable question(s). I'm not sure I follow your point here. Provability and undecidability are areas of Metalogic that I have studied, if that's what you are getting at. Hope that helps, Jonathan Hoyle Eastman Kodak
From: Richard Harter on 30 Nov 2005 01:07 On 27 Nov 2005 18:33:15 -0800, "Jonathan Hoyle" <jonhoyle(a)mac.com> wrote: >In ZF Set Theory with no urelements, sets contain only other sets and >nothing else. The set of natural numbers, or example, is just a set of >sets. > >If you mean *all sets* then ZF and ZFC do not allow for the set of all >sets. This must be the case, since if the collection of all sets were >a set, it must therefore contain itself as a member, which violates the >Axiom of Foundation (AF), one of the axioms of ZF. Just as a side note, the axiom of foundation was not part of the original ZF and need not be included in an axiomization of ZF. The real killer is the axiom of replacement (or in the original Z the axiom of subsets (ausonderung)). It takes directly from V to Russell's paradox. > >In a conservative extension of ZFC called NBG >(vonNeumann-Bernays-Godel), you can have a collection of all sets, but >this collection is not a set, but rather a proper class, for the >reasons stated above. (All sets are classes, and those classes which >are not sets are called proper classes). NBG is equi-consistent to >ZFC; that is to say, if NBG is inconsistent, then it is only because >ZFC is as well. > >To have "the set of all sets", you must work within a set theory which >does not contain AF. These are called "non-Well Founded" Set Theories, >and sets which violates AF are called "non-well founded sets". >Obviously non-well founded sets in these theories do not exist in ZF, >and thus you cannot assume traditional ZF properties to them. > >Hope that helps, > >Jonathan Hoyle > Richard Harter, cri(a)tiac.net http://home.tiac.net/~cri, http://www.varinoma.com I started out in life with nothing. I still have most of it left.
From: zuhair on 10 Dec 2005 12:44 Dear Jonathan Hoyle: In a previous message you said that a set is a collection which follows ZFC axioms. But ZFC axioms speakes of sets. For example: axiom of extensionality states that:-Two sets are the same if and only if they have the same elements. Don't you think that there is something cyclical in what you are saying. Zuhair
From: Lee Rudolph on 10 Dec 2005 12:47
"zuhair" <zaljohar(a)yahoo.com> writes: >Dear Jonathan Hoyle: > >In a previous message you said that a set >is a collection which follows ZFC axioms. > >But ZFC axioms speakes of sets. > >For example: axiom of extensionality >states that:-Two sets are the same if and only if they have the same >elements. > > >Don't you think that there is something cyclical >in what you are saying. Accepting your implicit statement as true for the sake of argument--so what? Not all circles are vicious. Lee Rudolph |