The set of All sets
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From: Ross A. Finlayson on 27 Nov 2005 23:20 Jonathan Hoyle 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. > > 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 What's the class of all classes, in lieu of a set of all sets? Answer: there is none. Proper classes aren't allowed to contain proper classes. The order type of ordinals would still be an ordinal, in NBG it's not a set, but it still contains the structure of an ordinal. That the order type of ordinals would be an ordinal is called the Burali-Forti paradox for Cesare Burali-Forti. For some, having classes, not classes at school but these contents of classified collections, does not seem to be resolution of the problems of unrestricted comprehension, which in a sense is logical induction. The set surcease, class conundrum, group game, shell shuffle, doesn't have a universe. To get to talking about a universe, there are some difficulties. Ross
From: zuhair on 28 Nov 2005 00:17 Jonathan Hoyle 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, Define " collection " Define set Zuhair.......... it must therefore contain itself as a member, which violates the > Axiom of Foundation (AF), one of the axioms of ZF. > > 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
From: Jonathan Hoyle on 28 Nov 2005 14:36 >> Define " collection " >> Define set Sets are defined within the given theory. In ZFC, there are axioms to tell you what sets are. I use the term "collection" to describe those things which may be modelled as a set in one theory, but not necessarily in another. For example, the "collection of all sets" would not be a set in ZFC, would be a proper class in NBG, and would be a full-fledged set in some non-WF theories. Of course this assumes you are working within a framework which can be simultaneously modelled by different set theories. Hope that helps, Jonathan Hoyle Eastman Kodak
From: zuhair on 29 Nov 2005 07:38 Jonathan Hoyle wrote: > >> Define " collection " > >> Define set > > Sets are defined within the given theory. In ZFC, there are axioms to > tell you what sets are. > > I use the term "collection" to describe those things which may be > modelled as a set in one theory, but not necessarily in another. For > example, the "collection of all sets" would not be a set in ZFC, would > be a proper class in NBG, and would be a full-fledged set in some > non-WF theories. > > Of course this assumes you are working within a framework which can be > simultaneously modelled by different set theories. And what is that framwork. 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. you should define collection in a clear manner. Zuhair > > Hope that helps, > > Jonathan Hoyle > Eastman Kodak
From: Jonathan Hoyle on 29 Nov 2005 11:02
>> 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 |