The size of proper classes?
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From: zuhair on 2 Dec 2009 22:56 On Dec 2, 8:58 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Dec 2, 1:35 pm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote: > > > The strongest class form of the Axiom of > > Choice has all proper classes equinumerous to the class of > > all ordinal numbers. > > What about the just-plain set/class distinction ITSELF, > in the form of a limitation-of-size principle, i.e., the claim > (or definition, even) that a class is proper if and only if it is > "equinumerous" to the class of all sets? And Is Therefore > "supernumerous" to each individual set? Is *that* involved with > or dependent on Choice? Not it imply choice, i.e Choice would be a theorem, actually global choice. How I see matters, is that there is no difference between saying that "x is a proper class iff x is equinumerous with the class of all sets", and between saying "x is a proper class iff x is equinumerous with the class of all ordinal numbers (here ordinal numbers mean ordinals that are sets)", I am speaking of course about NBG\MK approach. However perhaps I am mistaken. Zuhair
From: Aatu Koskensilta on 3 Dec 2009 08:58 zuhair <zaljohar(a)gmail.com> writes: > How I see matters, is that there is no difference between saying that > "x is a proper class iff x is equinumerous with the class of all > sets", and between saying "x is a proper class iff x is equinumerous > with the class of all ordinal numbers (here ordinal numbers mean > ordinals that are sets)", I am speaking of course about NBG\MK > approach. However perhaps I am mistaken. You're not. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 3 Dec 2009 11:49 zuhair <zaljohar(a)gmail.com> writes: > If we take ZF + inaccessibles , then these inaccessible sets are > "sets" i.e. are members of other classes, but yet they have the size > of a proper class in NBG\MK with no inaccessibles. This explanation or claim doesn't really make much sense. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Herman Rubin on 3 Dec 2009 12:00 In article <d2a856ff-ae3a-4dbd-9799-f3a5637d5245(a)e31g2000vbm.googlegroups.com>, zuhair <zaljohar(a)gmail.com> wrote: >On Dec 2, 1:35=A0pm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote: >> In article <b1d5c9ac-d3ca-42a2-9d9c-e2decdf0c...(a)m35g2000vbi.googlegroups= >.com>, >> zuhair =A0<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. =A0The strongest class form of the Axiom of >> Choice has all proper classes equinumerous to the class of >> all ordinal numbers. =A0See the book _Equivalents of the >> Axiom of Choice II_ by Herman Rubin and Jean E. Rubin. =A0The >> 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 There exist models where all proper classes have the same cardinality; i.e., the universe is equinumerous with the class of ordinal numbers. There exist models where there are proper classes which are neither larger nor smaller than the class of all ordinal numbers. I am almost certain that there exist models with proper classes strictly larger than the class of all ordinal numbers, and all comparable. -- 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: Aatu Koskensilta on 3 Dec 2009 12:16
hrubin(a)odds.stat.purdue.edu (Herman Rubin) writes: > I am almost certain that there exist models with proper classes > strictly larger than the class of all ordinal numbers, and all > comparable. Using the consistency of ZFC + "there is an inaccessible" (and standard well known independence results) it's easy to show there are such models. As noted, by some tweaking we can remove the inessential inaccessible, bringing us back to ZFC and MK in terms of consistency strength. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |