Recursive Cardinals.
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From: zuhair on 28 Dec 2009 14:51 On Dec 28, 7:22 am, David Hartley <m...(a)privacy.net> wrote: > In message > <88909956-acd2-407d-a673-c42065e5a...(a)k19g2000yqc.googlegroups.com>, > zuhair <zaljo...(a)gmail.com> writes > > >Now how much countable bracket sequences we can have? > > the answer is: we can have Power(omega) of these countable bracket > >sequences ONLY. > > How do you distinguish between the sets with the bracket sequences > <1, 2, 3, ...> and <0, 2, 3, ...>? Are they not the same by > Extensionality? Yea, in reality this is a difficult question, if you read what David was speaking about different models, then we can actually have them stand for different sets, however I was of the same opinion of yours really, but this needs to be better defined. Thanks Zuhair > > -- > David Hartley
From: zuhair on 28 Dec 2009 16:03 In my reply to David Libert above I mentioned that working from outside should not come in conflict with inside, and since David gave nice simple examples, so let me clarify this point using his own examples: Lets take identity theory itself, Lets add to it the following axiom: For all x,y ( x=y ) Now we cannot work from outside and bring a model which has Exist x ,y ( ~x=y ) , This would be in contradiction to working from inside. So all models of this theory must not include distinct objects. This matter is somewhat similar to the singleton towers, the lemma that I mentioned would prevent us from having proper class of singleton towers in *ALL* models of ZF-Reg. However I might be mistaken of course. I'll see what David Libert would come up with. Zuhair
From: David Libert on 29 Dec 2009 05:45 zuhair (zaljohar(a)gmail.com) writes: > In my reply to David Libert above I mentioned that working from > outside should not come in conflict with inside, and since David gave > nice simple examples, so let me clarify this point using his own > examples: > > Lets take identity theory itself, > > Lets add to it the following axiom: > > For all x,y ( x=y ) > > Now we cannot work from outside and bring a model which has > > Exist x ,y ( ~x=y ) , > > This would be in contradiction to working from inside. Ok, I think I see what you are doing now. In this identity theory example, you are saying consider instead of the pure theory of identity, as I said earlier, it supplimented by the extra axiom For all x,y (x=y). In your discussions in previous articles about recursive cardinals at one point you had mentioned an axiom: for every x there is an x-recursive set. And more recently you mentioned the axiom you had been thinking of: >I was obviously working with an axiom in my mind that I myself didn't >know about , and that is: > >For any two singleton towers x,y ( x e TC(y) or y e TC(x) ). > >This will cut down the possibility of singleton towers having disjoint >transitive closures. That quoted from your Dec 27 article in this thread. So I will write about what happens when we add these axioms and similar ones to ZF - regularity. First I quote the rest of your parent article: > So all models of this theory must not include distinct objects. > > This matter is somewhat similar to the singleton towers, the lemma > that I mentioned would prevent us from having proper class of > singleton towers in *ALL* models of ZF-Reg. > > However I might be mistaken of course. > > I'll see what David Libert would come up with. > > Zuhair So the question is what happens with the class of x-recursive sets for various x in various models of ZF - regularity. For general ZF - regularity models and various x there are two issues: is the class of all x-recursive sets empty and is the class of all x-recursive sets a proper class. If either of these happen for a particular x it makes problems for defining cardinality of x using x-recursice sets. On the other hand, if neither of those happen, if that class is a non-empty set, then we can make a good definition of cardinality as the set of all x-recursive sets. In general models of ZF - regularity either of those cases is possible. I had in my first article about cardinality being undefinable written about getting a proper class of doubleton towers and no singelton towers. Along similar lines, the question about size of the class of x-recursice sets as x varies: can be in any of the 3 cases empty, non-empty set and proper class, in any simulataeous pattern as x varies. But as you say, we can also consider variants of ZF - refularity by adding extra axioms, maybe axiomatizing those cases away. So one of your recent;ly suggested axioms was to axiomatize away the emoty case: aciomatize that the class is always non-empty. So that does get rid of one problem for the definition. You could also just axiomatize the other side too. I think you did this in a recent article. Just axiomatize that that class is for all x a set. Or state them together as I think you did: axiomatize that it is a non-empty set. Doing that, you do get a good definition of cardality as the set of all x-recursive sets, So of course, if my models with cardinality are correct they don't satisfy these extra axioms. Let's consider instead your other recent axiom as I quoted above: >For any two singleton towers x,y ( x e TC(y) or y e TC(x) ). In the singleton towers case as you explcitly said, this does imply there are only countably many recursive singletons as you have been saying. So if you axiomatize that the class is non-empty and also this comparabilty axiom above, you get a good definition of cardinality for #x = 1. There is the possibilty of generalizing your definition to all x, ie doubleton towers etc. I think I have come up with models similar to my previous ones, using similar methods. First, there can be a model where for every x the class is a non-empty set. So this would be a model where your definition of cardinality does work well for all x. But I think I can also come up with models of the analogue of your new comparability axioms above for other cardinalities, ie x of other sizes, where there is still a proper class of x-recursive sets. These are for #x > 1. So this comaparability axiom doesn't seem enough. But if you want to axiomatize that all those classes are non-empty sets directly, then it is ok for definining cardinality. -- David Libert ah170(a)FreeNet.Carleton.CA
From: zuhair on 29 Dec 2009 07:21 On Dec 29, 5:45 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > zuhair (zaljo...(a)gmail.com) writes: > > In my reply to David Libert above I mentioned that working from > > outside should not come in conflict with inside, and since David gave > > nice simple examples, so let me clarify this point using his own > > examples: > > > Lets take identity theory itself, > > > Lets add to it the following axiom: > > > For all x,y ( x=y ) > > > Now we cannot work from outside and bring a model which has > > > Exist x ,y ( ~x=y ) , > > > This would be in contradiction to working from inside. > > Ok, I think I see what you are doing now. > > In this identity theory example, you are saying consider instead > of the pure theory of identity, as I said earlier, it supplimented > by the extra axiom For all x,y (x=y). > > In your discussions in previous articles about recursive cardinals > at one point you had mentioned an axiom: for every x there is > an x-recursive set. > > And more recently you mentioned the axiom you had been thinking > of: > > >I was obviously working with an axiom in my mind that I myself didn't > >know about , and that is: > > >For any two singleton towers x,y ( x e TC(y) or y e TC(x) ). > > >This will cut down the possibility of singleton towers having disjoint > >transitive closures. > > That quoted from your Dec 27 article in this thread. > > So I will write about what happens when we add these axioms and > similar ones to ZF - regularity. > > First I quote the rest of your parent article: > > > So all models of this theory must not include distinct objects. > > > This matter is somewhat similar to the singleton towers, the lemma > > that I mentioned would prevent us from having proper class of > > singleton towers in *ALL* models of ZF-Reg. > > > However I might be mistaken of course. > > > I'll see what David Libert would come up with. > > > Zuhair > > So the question is what happens with the class of x-recursive > sets for various x in various models of ZF - regularity. > > For general ZF - regularity models and various x there are > two issues: is the class of all x-recursive sets empty > and is the class of all x-recursive sets a proper class. > > If either of these happen for a particular x it makes problems > for defining cardinality of x using x-recursice sets. > > On the other hand, if neither of those happen, if that class > is a non-empty set, then we can make a good definition of > cardinality as the set of all x-recursive sets. > > In general models of ZF - regularity either of those cases > is possible. > > I had in my first article about cardinality being undefinable > written about getting a proper class of doubleton towers and > no singelton towers. > > Along similar lines, the question about size of the class > of x-recursice sets as x varies: can be in any of the 3 cases > empty, non-empty set and proper class, in any simulataeous > pattern as x varies. > > But as you say, we can also consider variants of > ZF - refularity by adding extra axioms, maybe axiomatizing those > cases away. > > So one of your recent;ly suggested axioms was to axiomatize > away the emoty case: aciomatize that the class is always > non-empty. So that does get rid of one problem for the > definition. > > You could also just axiomatize the other side too. I think > you did this in a recent article. > > Just axiomatize that that class is for all x a set. > > Or state them together as I think you did: axiomatize that > it is a non-empty set. > > Doing that, you do get a good definition of cardality > as the set of all x-recursive sets, > > So of course, if my models with cardinality are correct > they don't satisfy these extra axioms. > > Let's consider instead your other recent axiom as I quoted > above: > > >For any two singleton towers x,y ( x e TC(y) or y e TC(x) ). > > In the singleton towers case as you explcitly said, this > does imply there are only countably many recursive singletons > as you have been saying. > > So if you axiomatize that the class is non-empty and also > this comparabilty axiom above, you get a good definition > of cardinality for #x = 1. > > There is the possibilty of generalizing your definition > to all x, ie doubleton towers etc. > > I think I have come up with models similar to my previous > ones, using similar methods. > > First, there can be a model where for every x the class > is a non-empty set. So this would be a model where > your definition of cardinality does work well for all x. No I am coming to think that this is impossible, because this will exhaust the bracket sequences,especially if x is supernumerous to power (omega). > > But I think I can also come up with models of the > analogue of your new comparability axioms above for > other cardinalities, ie x of other sizes, where > there is still a proper class of x-recursive sets. > These are for #x > 1. > > So this comaparability axiom doesn't seem enough. > > But if you want to axiomatize that all those classes > are non-empty sets directly, then it is ok for > definining cardinality. > > -- > David Libert ah...(a)FreeNet.Carleton.CA No David I think you didn't catch my basic argument in my latest reply to you. Even without this axiom that I've wrote lately, I still think that you cannot have more than Power(omega) of singleton towers. I shall quote again: This is the basic issue: --Quote-- However the matter is actually deeper than that! Even if I suppose the existence of singleton towers that are disjoint at their transitive closures level, i.e. having disjoint transitive closures, even if we assume that, don't forget that the identity of every singleton tower would depend on the bracket sequence of it, and you know that from the lemma that I have mentioned, you are only permitted to have a *countable* bracket sequence for each singleton tower! Now how much countable bracket sequences we can have? the answer is: we can have Power(omega) of these countable bracket sequences ONLY. So EVEN if you have singleton towers with disjoint transitive closures still, you cannot have more than Power(omega) of them, because each singleton tower must correspond to a countable bracket sequence. All of that is a consequence of the *Lemma* that I mentioned which is a LIMITING factor, it limits you from being able to stipulate that you can have a proper classes of them at will, like the case with Ur- elements and Quine atoms, or the CIRCULAR singleton towers (a Quine atom is actually a circular singleton tower of the first degree). So ONLY What I call as circular sets or Ur-elements can be proliferated *at will* to the size of proper classes. The net result is that: you CANNOT have a model of ZF-Reg. with a proper class of these towers even if you do it from outside, because the lemma from inside will contradict this. If you work from outside, then this should not contradict matters from inside, lest you will end up with a sort of non-standard singleton towers, as we have non-standard naturals, and so on. But we don't want that! aren't we. If you want to work from outside, then you should work in such a manner that the lemma that I mentioned do not contradict with your models, your work from outside must be one that we can test if it contradict the lemma that I've mentioned or not? If we cannot show how your work form outside is tested by the lemma I've mentioned, then your work cannot be tested to know if it contradict this lemma or not, then we cannot be sure of your models, aren't we. --Quote finished--- So the basic issue is actually not related to these additional axioms of mine you were talking about, it is related to the very basic concept of having a proper class of singleton towers, which I still don't see how can it be done EVEN from outside EVEN in absence of these additional axioms. I spoke about the resulting inevitable conflict between "outside" and "inside". The *lemma* and not the axioms would be the "LIMITING" factor on *all* models from outside that define a proper class of singleton towers, so it will contradict all of them. So still we are differing about the same particular matter, I am saying we cannot have a proper class of singleton towers in all models of ZF-Reg., (unless these are non standard singleton towers, like how we have non standard naturals), and you are saying that we do have this proper class. Zuhair
From: David Libert on 31 Dec 2009 04:27
Correcting references: David Libert (ah170(a)FreeNet.Carleton.CA) writes: [Deletion] > In > > [1] David Libert "A new definition of Cardinality" > sci.logic, sci.math Nov 23, 2009 > http://groups.google.com/group/sci.math/msg/721cb8170033cf84 > > I constructed a ZFC - regularity model with a proper class of > recursive cardinals. > > I still think that construction is ok. [Deletion] > I will close with some references about my models. > > I have had some other models besides [1]. > > > These and [1] and background were referenced in > > [2] David Libert " Extensionality and Circular objects" > sci.logic, sci.math Dec 23, 2009 > http://groups.google.com/group/sci.logic/msg/f3ed79cb6bf9fb5e > > > > I recently in this thread had those two followup articles on points > about these conistructions: > > > [3] David Libert "Recursive Cardinals" > sci.logic, sci.math Dec 26, 2009 > http://groups.google.com/group/sci.logic/msg/2a5d12e702092b36 > > > > [3] David Libert "Recursive Cardinals" > sci.logic, sci.math Dec 29, 2009 > http://groups.google.com/group/sci.logic/msg/a1b67d758208e941 > > > In [3] I discussed how these constructions use atoms toward defining > sets in the final model. > > In [4] I noted how and why these arguments work both from inside > and outside the constructed models. > > > -- > David Libert ah170(a)FreeNet.Carleton.CA The article about the atoms was actually [5] David Libert " Extensionality and Circular objects" sci.logic, sci.math Dec 23, 2009 http://groups.google.com/group/sci.logic/msg/ecfd5cc4f0f3a072 The article about inside and outside the model was the first [3] above. (I see when I cut and pasted I forgot to change [3] to [4]). The second [3] above = [4] was about adding other axioms to ZF - regularity. -- David Libert ah170(a)FreeNet.Carleton.CA |