The size of proper classes?
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From: zuhair on 3 Dec 2009 20:44 On Dec 3, 12:00 pm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote: > In article <d2a856ff-ae3a-4dbd-9799-f3a5637d5...(a)e31g2000vbm.googlegroups..com>, > > > > > > zuhair <zaljo...(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. Yes, it is these later models that I am seeking, Can you please specify at least one of them or point to a reference about these models. In a separate topic of this Usenet, I recently spoke about Z+size limitation, I think this would be an example of such a theory. The importance of that issue is connected to the proof that the cardinals that I defined would be sets! they are superior to the ordinary cardinals in that they do not require choice like Von Neumann Cardinals and in that they do not always require regularity as Scott's cardinals. Cardinality(x) is the equivalence class of all sets having every member of their transitive closures strictly subnumerous to x, under equivalence relation "bijection". 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 3 Dec 2009 20:58 On Dec 3, 12:16 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > hru...(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. Ok, then, that is important. Then this mean that adding the following size limitation schema to Z set theory, bears no problem. Axiom schema of size limitation: 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 axioms. > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: zuhair on 3 Dec 2009 21:00 On Dec 3, 11:49 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > zuhair <zaljo...(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. Agreed. Zuhair > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: David Libert on 3 Dec 2009 23:05 zuhair (zaljohar(a)gmail.com) writes: [Deletion] > In a separate topic of this Usenet, I recently > spoke about Z+size limitation, I think this > would be an example of such a theory. > > The importance of that issue is connected > to the proof that the cardinals that I defined > would be sets! they are superior to the ordinary > cardinals in that they do not require choice > like Von Neumann Cardinals > and in that they do not always require regularity > as Scott's cardinals. You and I each posted articles models of ZFC without regularity, in which there are a proper class of sets with all members of the traansitive closure being singletons. My article was [1] David Libert "A new definition of Cardinality" sci.logic, sci.math Nov 23 http://groups.google.com/group/sci.math/msg/721cb8170033cf84 Yours was [2] Zuhair "A new definition of Cardinality" sci.logic, sci.math Nov 25 http://groups.google.com/group/sci.math/msg/68fa234768c92dcc Your Z+size limitation is so similar to ZF insofar as I have unde4rstood it, I think similar models would apply if the basic theory is ok. So these seem to show me that regularity is required. > Cardinality(x) is the equivalence class of all sets > having every member of their transitive closures > strictly subnumerous to x, under equivalence > relation "bijection". > > Zuhair This is the same as your first definition, which was given in the thread of [1] & [2]. I discussed this and also your second definition in [3] David Libert "The magic of Hereditarily Hereditary Cardinals" sci.logic, sci.math Nov 29 http://groups.google.com/group/sci.math/msg/1b40b261aeff6e96 In [3] I defined signatures for such definitions. The correct signature for your definition above is = < . In [3] I incorrectly wrote this as <= < , and noted that correction in a followup to [3]. So regarding = < definition as you have above, in [3] I argued that for x = A amorphous, Cardinality(x) = {} . I still think that was correct. In [3] I went on to note Cardinality({}) = {} . I now think this was wrong. I think Cardinality({}) = {{}} . So the first overlap I noted of A and {} getting same Cardinality is retracted. But in [3] I also noted we could also get a different Cohen model with A1, A2 non-isomorphic amorphous sets, and so have Cardinality(A1) = {} Cardinality(A2) = {} still making a problem for this definition. I still think that last point is correct, making a problem for this definition. -- David Libert ah170(a)FreeNet.Carleton.CA
From: zuhair on 4 Dec 2009 07:29
On Dec 3, 11:05 pm, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > zuhair (zaljo...(a)gmail.com) writes: > Let me clarify matters: This definition of Cardinality which is the same one I posted in topic "a new definition of Cardinality", works with Regularity OR Choice. So if we don't have Regularity but we have Choice then the definition work! and also if we don't have Choice but we have Regularity also the definition work, however the definition fail if we neither have Regularity nor Choice, i.e. it fails for example in "ZF minus Regularity" and it is this later condition that you are speaking of, which is true. So in ZF, the definition work in ZFC minus Regularity, the definition work in ZFC, the definition work in ZF minus Regularity, the definition fail. I am not sure if Scott Cardinals have the same property, they might! but I got they idea that they don't work outside Regularity , anyhow , but this definition is simpler anyway. I hope I clarified this point. Zuhair |