The size of proper classes?
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From: zuhair on 2 Dec 2009 07:53 Working in NBG\MK minus choice Can there exist a proper class x that is not supernumerous to the class of all ordinals that are sets? x supernumerous to y <-> Exist f (f:y-->x, f is injective) I always had the idea that the class of all ordinals that are sets, is the smallest proper class, i.e. there do not exist a proper class that is strictly subnumerous to it, but can there exist a proper class that is incomparable to it, i.e. there do not exist any injection between it and that proper class. If so can one give an example of such a proper class? Zuhair
From: Herman Rubin on 2 Dec 2009 13:35 In article <b1d5c9ac-d3ca-42a2-9d9c-e2decdf0c3b2(a)m35g2000vbi.googlegroups.com>, zuhair <zaljohar(a)gmail.com> wrote: >Working in NBG\MK minus choice > Can there exist a proper class x that is not supernumerous to the >class of all ordinals that are sets? >x supernumerous to y <-> Exist f (f:y-->x, f is injective) There are certainly Fraenkel-Mostowski models in which this is false, and I believe Cohen models as well. Fraenkel-Mostowski models are not models of ZF, but of ZFU; the models needed are models of NBG, but Fraenkel-Mostowski models can be extended. >I always had the idea that the class of all ordinals that are sets, is >the smallest proper class, i.e. there do not exist a proper class that >is strictly subnumerous to it, but can there exist a proper class that >is incomparable to it, i.e. there do not exist any injection between >it and that proper class. >If so can one give an example of such a proper class? Not necessarily. The strongest class form of the Axiom of Choice has all proper classes equinumerous to the class of all ordinal numbers. See the book _Equivalents of the Axiom of Choice II_ by Herman Rubin and Jean E. Rubin. The construction in Godel's book, _Consistencey of the Continuum Hypothesis_, constructs and inner model of NBG in which it is true that the class of ordinal numbers is equinumerous with the universe. -- 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
From: zuhair on 2 Dec 2009 15:27 On Dec 2, 1:35 pm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote: > In article <b1d5c9ac-d3ca-42a2-9d9c-e2decdf0c...(a)m35g2000vbi.googlegroups..com>, > > zuhair <zaljo...(a)gmail.com> wrote: > >Working in NBG\MK minus choice > > Can there exist a proper class x that is not supernumerous to the > >class of all ordinals that are sets? > >x supernumerous to y <-> Exist f (f:y-->x, f is injective) > > There are certainly Fraenkel-Mostowski models in which this > is false, and I believe Cohen models as well. > Fraenkel-Mostowski models are not models of ZF, but of ZFU; > the models needed are models of NBG, but Fraenkel-Mostowski > models can be extended. If I didn't misunderstand you, what you are saying is the following: There cannot exist a proper class x that is not supernumerous to the class of all ordinals, in other words what you are saying is: the Frankel-Mostowski models prove that every proper class is supernumerous to the class of all ordinals that are sets, i.e for any class x to be a proper class then there must exist an injection from the class of all ordinals that are sets to the class x. Is that what you are saying? > > >I always had the idea that the class of all ordinals that are sets, is > >the smallest proper class, i.e. there do not exist a proper class that > >is strictly subnumerous to it, but can there exist a proper class that > >is incomparable to it, i.e. there do not exist any injection between > >it and that proper class. > >If so can one give an example of such a proper class? > > Not necessarily. The strongest class form of the Axiom of > Choice has all proper classes equinumerous to the class of > all ordinal numbers. See the book _Equivalents of the > Axiom of Choice II_ by Herman Rubin and Jean E. Rubin. The > construction in Godel's book, _Consistencey of the > Continuum Hypothesis_, constructs and inner model of NBG in > which it is true that the class of ordinal numbers is > equinumerous with the universe. This is a little bit vague, what was you referring to when you said "Not necessarily"?Did you mean that we can have a proper class that is strictly subnumerous to the class of all ordinals that are sets? or can there exist a proper class that is not comparable to the class of all ordinals that are sets? these points are not clear from your answer. Thanks for the references. Zuhair > > -- > 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 > hru...(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
From: zuhair on 2 Dec 2009 16:09 On Dec 2, 1:35 pm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote: > In article <b1d5c9ac-d3ca-42a2-9d9c-e2decdf0c...(a)m35g2000vbi.googlegroups..com>, > > zuhair <zaljo...(a)gmail.com> wrote: > >Working in NBG\MK minus choice > > Can there exist a proper class x that is not supernumerous to the > >class of all ordinals that are sets? > >x supernumerous to y <-> Exist f (f:y-->x, f is injective) > > There are certainly Fraenkel-Mostowski models in which this > is false, and I believe Cohen models as well. > Fraenkel-Mostowski models are not models of ZF, but of ZFU; > the models needed are models of NBG, but Fraenkel-Mostowski > models can be extended. > > >I always had the idea that the class of all ordinals that are sets, is > >the smallest proper class, i.e. there do not exist a proper class that > >is strictly subnumerous to it, but can there exist a proper class that > >is incomparable to it, i.e. there do not exist any injection between > >it and that proper class. > >If so can one give an example of such a proper class? > > Not necessarily. The strongest class form of the Axiom of > Choice has all proper classes equinumerous to the class of > all ordinal numbers. See the book _Equivalents of the > Axiom of Choice II_ by Herman Rubin and Jean E. Rubin. The > construction in Godel's book, _Consistencey of the > Continuum Hypothesis_, constructs and inner model of NBG in > which it is true that the class of ordinal numbers is > equinumerous with the universe. > > -- > 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 > hru...(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 It seems that I was not clear in asking my question. Let me re ask my question using more precise terminology: Is the following a theorem schema of ZF(without choice)? If phi(y) is a formula in which at least y is free, and in which x is not free, then all closures of ~ for all d ( d is ordinal -> Exist x ( for all y (y e x -> phi(y)) and d equinumerous to x ) ) -> Exist x for all y ( y e x <-> phi(y) ) are theorems. I think the idea behind the question is very clear, if we cannot put all ordinals that are sets into one-one relation with sets fulfilling the predicate phi (this is equivalent to saying that we cannot have an injection from the class of all ordinals that are sets to the class of all sets fulfilling the predicate phi), then the predicate phi defines a set, i.e. the class of exactly all sets for which the predicate phi holds is a set. Now is that true in ZF(without choice) ? is that true in ZF without choice and without regularity? I hope my question is clear this time? Zuhair
From: zuhair on 2 Dec 2009 16:13
On Dec 2, 1:35 pm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote: > In article <b1d5c9ac-d3ca-42a2-9d9c-e2decdf0c...(a)m35g2000vbi.googlegroups..com>, > > zuhair <zaljo...(a)gmail.com> wrote: > >Working in NBG\MK minus choice > > Can there exist a proper class x that is not supernumerous to the > >class of all ordinals that are sets? > >x supernumerous to y <-> Exist f (f:y-->x, f is injective) > > There are certainly Fraenkel-Mostowski models in which this > is false, and I believe Cohen models as well. > Fraenkel-Mostowski models are not models of ZF, but of ZFU; > the models needed are models of NBG, but Fraenkel-Mostowski > models can be extended. > > >I always had the idea that the class of all ordinals that are sets, is > >the smallest proper class, i.e. there do not exist a proper class that > >is strictly subnumerous to it, but can there exist a proper class that > >is incomparable to it, i.e. there do not exist any injection between > >it and that proper class. > >If so can one give an example of such a proper class? > > Not necessarily. The strongest class form of the Axiom of > Choice has all proper classes equinumerous to the class of > all ordinal numbers. See the book _Equivalents of the > Axiom of Choice II_ by Herman Rubin and Jean E. Rubin. The > construction in Godel's book, _Consistencey of the > Continuum Hypothesis_, constructs and inner model of NBG in > which it is true that the class of ordinal numbers is > equinumerous with the universe. > > -- > 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 > hru...(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 It seems that I was not clear in asking my question. Let me re ask my question using more precise terminology: Is the following a theorem schema of ZF(without choice)? If phi(y) is a formula in which at least y is free, and in which x is not free, then all closures of ~ for all d ( d is ordinal -> Exist x ( for all y ( y e x -> phi(y) ) and d equinumerous to x ) ) -> Exist x for all y ( y e x <-> phi(y) ) are theorems. I think the idea behind the question is very clear, if we cannot put all ordinals that are sets into one-one relation with sets fulfilling the predicate phi (this is equivalent to saying that we cannot have an injection from the class of all ordinals that are sets to the class of all sets fulfilling the predicate phi), then the predicate phi defines a set, i.e. the class of exactly all sets for which the predicate phi holds is a set. Now is that true in ZF(without choice) ? is that true in ZF without choice and without regularity? I hope my question is clear this time? Zuhair |