The size of proper classes?
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From: George Greene on 2 Dec 2009 21:03 On Dec 2, 7:53 am, 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? Can there be proper-class ordinals?? "all ordinals that are sets" is almost surely redundant -- ordinals HAVE to be sets! Proper classes, in ADDITION to being "big enough", ALSO have to NOT be members of other classes! Every ordinal is a member of its successor class, so it cannot be a proper class.
From: zuhair on 2 Dec 2009 21:16 On Dec 2, 8:40 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Dec 2, 7:53 am, 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? > > This is NOT a well-formed question. > While proper classes may be first-class objects in NBG, > they are not so in ZF or other more usual theories. > Therefore, EVEN ASKING whether they are or aren't "supernumerous" > is problematic, INside the theory, since INside the theory, they are > not > even in the domain of discourse. That is not correct. Proper classes exist in the domain of discourse of NBG and MK, and we can even define cardinalities of these proper classes, the relation "supernumerous" is well defined for proper classes, and there is not problem with that whatsoever, so your remarks above are all mistaken, when you are speaking of NBG\MK. However they correct when you are speaking about theories that permits 'sets' only, i.e. non ur-elements(classes) that are members of other classes. so in these theories yes you are correct those objects are not in the domain of discourse, but yet we can have alternative methods of dealing with them (please see my second reply to Herman Rubin, in order to have an example of such equivalent treatements in ZF and similar "set" theories). These > > Even when they are in the domain, as in NBG, the set-class > distinction is sharp AND SIMPLE, which is why long replies to this > (even when they are expert and correct) are STILL misguided. > You need to know the FUNDAMENTAL point about sizes of > proper classes: The size of EVERY proper class is the SAME size > as the size of the class of all sets. No that is not correct. Not all models of NBG stipulate that, you are taking about a model of NBG\MK that allows global choice, that is not the only model, there are other models in which proper classes can be defined as being supernumerous to the class of all ordinals that are sets which is the smallest possible proper class. If you read my post carefully I said NBG(minus choice) , what I mean by that NBG that do not contain what you've just mentioned (your are talking about NBG\MK in which the size limitation axiom stipulate that a set is a class that is strictly subnumerous to V (the class of all sets)) there are many variants of this size limitation axioms, some of them even don't mention V at all. In a model of NBG\MK minus choice, you cannot have all proper classes equal in size (i.e.equinumerous), but in any model of NBG\MK the least (in size) possible proper class is the class of all ordinals that are sets, so you can stipulate that a set is a class that is not supernumerous (equal or bigger) than the class of all ordinals that are sets, and in this way you leave room for V to be bigger (strictly supernumerous) to the class of all ordinals that are sets, thus choice would not be entailed in this model, there are various other treatments of "size limitation" axiom that allow different comparisons than those that I have mentioned. > THAT IS the set/class distinction in these contexts: "subnumerous to" > vs. "equinumerous to" the class of all sets. No not necessarily, you are talking about one particular model of NBG \MK that entail global choice, which is not the only model, actually this model is too strong to the degree of making it undesirable, we need the weaker models not this strong model. > > I can see a motivation for your question coming out of the fact that > it > initially appears that "most" sets are NOT ordinals, yet the class of > all > ordinals cannot be a set. So it is perhaps counter-intuitive that > something > consisting of as small and special a sliver of the class of all sets > as JUST > the ordinals could be the same size as the class of all sets. > Especially since, without choice, you may not be able to show the > existence of the actual bijection confirming the equipollence. > But then again, you may, in SOME contexts anyway. Yes, that is correct somehow. As I said above not in all models of NBG \MK you have the size of the class of all sets (usually denoted as V) equal to the size of the class of all ordinals that are sets, as I said this concept is too strong. The main idea is actually the following: a) the class of all ordinals that are sets is the smallest possible proper class in all models of NBG\MK b) What is strictly subnumerous to the class of all ordinals that are sets, is a set and what is not comparable to the class of all ordinals that are sets, is a set. I am sure of the first half of b), but I am not sure of the second statement in b). Regards Zuhair
From: George Greene on 2 Dec 2009 22:06 On Dec 2, 9:16 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Dec 2, 8:40 pm, George Greene <gree...(a)email.unc.edu> wrote: > > > On Dec 2, 7:53 am, 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? > > > This is NOT a well-formed question. You were right. I was wrong. I retracted the message. Unfortunately I was not in a position to retract it immediately and you rebutted before I could retract it.
From: Aatu Koskensilta on 2 Dec 2009 22:06 George Greene <greeneg(a)email.unc.edu> writes: > 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? This principle, claim, definition (obviously) implies global choice. Just pause to think about it for a moment, bearing in mind the class of ordinals is a proper class. If we're willing to take the consistency of ZFC + "there is an inaccessible" as given, it's easy to see neither NBG nor MK proves all proper classes are the same size. We need but recall that <V_kappa, V_kappa+1> is a model of MK for an inaccessible kappa, and that it's consistent with ZFC that GCH fails arbitrarily badly at kappa. With some tweaking (e.g. by considering variants of ZFC with a restricted powerset axiom) we can eliminate, in so far as consistency strength is considered, the inessential inaccessible. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: George Greene on 2 Dec 2009 22:12
On Dec 2, 9:16 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Dec 2, 8:40 pm, George Greene <gree...(a)email.unc.edu> wrote: Actually I DID retract this before you replied to it. I wrote it at 8:40 and got forcibly logged off the computer I wrote it on, as I was realizing that it was wrong. I had to go to another computer and retracted it around 8:55. But you had probably already started replying to it and finished drafting your reply around 9:15. Plus it takes a while for retractions to propagate, and not all servers honor them. I was really hoping I would get to it before you did. |