The size of proper classes?
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From: George Greene on 9 Dec 2009 00:00 On Dec 7, 1:28 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > You have been misled by Herman Rubin's comments. In all the theories > considered in these threads a class is proper iff it is not an element > of any class. In particular, there are no proper-class ordinals in your > sense, although there are well-orderings of the universe of order-type > greater than that of the class of ordinals. Thank you kindly for the clarification. Zuhair was wanting to define a proper-class ordinal as any transitive class of transitive sets. That would make the class of ordinals a proper-class ordinal, but it is hard to see how anything smaller than that would ever get to be a proper-class ordinal, since "smaller-than" FOR ordinals entails membership.
From: George Greene on 9 Dec 2009 00:03 On Dec 7, 7:13 am, zuhair <zaljo...(a)gmail.com> wrote: > in Ackermann's all > ordinals must have successors weather they are proper classes or sets, So in that case, you are using a definition of the set/class distinction that is something OTHER than "not being a member of another class" -- unless Ackermann's is using a different definition of successor. > while in NBG\MK all ordinals that are sets have successor, but there > exist only ONE proper class ordinal and that is > the class of all set ordinals, and it doesn't have any successor, and > yet it is ordinal! Thank you for the clarification. If it is the only one then that is a lot easier to tolerate. We would prefer talking about it as a unique and special thing to talking about the existence of proper-class-ordinalS, PLURAL (sincethere is only one).
From: zuhair on 9 Dec 2009 06:19 On Dec 9, 12:00 am, George Greene <gree...(a)email.unc.edu> wrote: > On Dec 7, 1:28 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > You have been misled by Herman Rubin's comments. In all the theories > > considered in these threads a class is proper iff it is not an element > > of any class. In particular, there are no proper-class ordinals in your > > sense, although there are well-orderings of the universe of order-type > > greater than that of the class of ordinals. > > Thank you kindly for the clarification. > Zuhair was wanting to define a proper-class ordinal as > any transitive class of transitive sets. That would make the > class of ordinals a proper-class ordinal, but it is hard to see how > anything smaller than that would ever get to be a proper-class > ordinal, > since "smaller-than" FOR ordinals entails membership. I Know you are replying to Aato, but I should clarify some matters. Let us be precise here so that we don't have any misunderstandings. First the defintion of "ordinal" any "ordinal" not only an ordinal that is a proper class, any ordinal, OK, is assuming Regularity is: An ordinal is a transtive *class* of transitive sets. This is proved to be equivalent to the standard definition of *ordinal* if as I said we assume Regularity. Without assuming Regularity this definition would be modified to An ordinal is a transtive *class* of transitive sets, in which every non empty subclass of it must contain a member that is disjoint of it. Now this is also proved to be equivalent to the standard definition of ordinal. Now if that class was a set, then we have a "set ordinal" If that class was a proper class, then we have a "proper class ordinal". The second point, You said: That would make the class of ordinals a proper-class ordinal That is not the precise statement, since we don't have a class of all ordinals, otherwise this will lead to Burali-Forti paradox. What we have is: The class of all set ordinals. or more clearily The class of all ordinals that are sets. And this class is a proper class, and it is an ordinal. and in NBG\MK it is the ONLY proper class ordinal, that's why I used the symbole D to symbolize it, although the standard symbol for it is "ORD". YOU said: but it is hard to see how anything smaller than that would ever get to be a proper-class ordinal, since "smaller-than" FOR ordinals entails membership I think what you mean by smaller is "strictly subnumerous", if so then you are correct, every ordinal that is strictly subnumerous to the proper class ordinal ORD is a set, there is no debate about that, and you are right when you said "smaller than" for ordinals entails membership, that is correct of course. Just to clarify points. Zuhair
From: zuhair on 9 Dec 2009 06:28 On Dec 9, 12:03 am, George Greene <gree...(a)email.unc.edu> wrote: > On Dec 7, 7:13 am, zuhair <zaljo...(a)gmail.com> wrote: > > > in Ackermann's all > > ordinals must have successors weather they are proper classes or sets, > > So in that case, you are using a definition of the set/class > distinction that > is something OTHER than "not being a member of another class" -- > unless > Ackermann's is using a different definition of successor. > > > while in NBG\MK all ordinals that are sets have successor, but there > > exist only ONE proper class ordinal and that is > > the class of all set ordinals, and it doesn't have any successor, and > > yet it is ordinal! > > Thank you for the clarification. > If it is the only one then that is a lot easier to tolerate. > We would prefer > talking about it as a unique and special thing > to > talking about the existence of proper-class-ordinalS, PLURAL > (sincethere is only one). Yes, in NBG\MK and I think also related models of ZF like ZFU and the alike, there would exist ONLY ONE proper class ordinal, and that is the class of all set ordinals, which is unique, and it is denoted by the symbol "ORD" although I like to use the symbol D for short. However in Ackermann's class theory, we do have multiple proper class ordinals, each one succceed the other, exactly as it with sets, but the definition of proper classes in Ackermann's is not the same as that in NBG\MK\ZF related models. In Ackermann's the definition of a proper class is: a class that is not a set. and a set is a primitive concept in Ackermann's class theory. However we mainly use NBG\MK\ZF related models, and in those theories we do only have ONE proper class ordinal. NOTE ( I am not sure if there are ZF related models were we have many proper class ordinals like the case with Ackermann's, but I greatly doubt that). Zuhair
From: Herman Rubin on 9 Dec 2009 12:28
In article <f8f5b867-fddd-4c83-bd15-857e3f7be2f2(a)k17g2000yqh.googlegroups.com>, George Greene <greeneg(a)email.unc.edu> wrote: >On Dec 7, 1:28=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> You have been misled by Herman Rubin's comments. In all the theories >> considered in these threads a class is proper iff it is not an element >> of any class. In particular, there are no proper-class ordinals in your >> sense, although there are well-orderings of the universe of order-type >> greater than that of the class of ordinals. <Thank you kindly for the clarification. <Zuhair was wanting to define a proper-class ordinal as <any transitive class of transitive sets. That would make the <class of ordinals a proper-class ordinal, but it is hard to see how <anything smaller than that would ever get to be a proper-class <ordinal, <since "smaller-than" FOR ordinals entails membership. Any subclass of an ordinal is order-isomorphic to an ordinal. Therefore, and proper class which is a subclass of the class of all ordinal numbers (ordinals which are sets) must be order-isomorphic to the class of all ordinals, and hence equinumerous. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |