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From: David Libert on 26 Dec 2009 02:33 zuhair (zaljohar(a)gmail.com) writes: > The following is a modified quote from another subject in reply to > David Libert, but I think it is better to be mentioned here. > > Here I have presented an example of these x-Recursive Cardinals > were x is singleton, It shows that we cannot have > uncountably many recursive singletons, at least at informal level, > here these recursive singletons shall be called "singleton towers": You are responding here to my claims to construct a model of ZF - regularity with a proper class of singleton towers. And my other models, or at least the last 2 about cardinality being undefinable, heavily rested on being able to make a proper class of singleton towers or something similar, as I wrote myself recently. You are skeptical about the possibility of a proper class of singleton towers (indeed even uncountably many singleton towers). So as you point out, if there is indeed a problem as you suspect about a proper class of singleton towers that would show something is wrong with my models, as you speculated recentlty in another thread. I agree with all that. If there is a problem with a proper class of singleton towers then there is a problem for my models. I still think a proper class of singleton towers is possible. I will begin by trying to write out more clearly why I think so. I know you write below into the quoted article a claimed proof there can only be countably many singleton towers. Below I will preserve the quote of that and give detailed answers to it. But to start I will give more background to all this and try to state my side more clearly. That will be a context to answering your proof below. I think a big point in all this is conceptual: what the proofs are trying to do, what they are really claiming and what are acceptable methods to be using in such proofs. So I will try to express those areas as I understand them for all this more clearly. I will leave the quote of your article below to respond to it in detail after my opening discussion. But to get started, I will jump ahead and quote your closing comments of the article, because I think they are important for this whole issue of what is being discussed, and what constitutes a proper proof. So the quote from late in your article: > I might be mistaken though, but I would like to know how can we have > an uncountable number of these singleton towers? and even more > how can we have a proper class of them? I want a proof from within > the model, and not from outside it. > > > Zuhair This point, am I to give a proof from within the model, or from outside, is this whole point I have about clarifying the overall discussion. So here we go. We have been interested in the theory ZF - regularity. You have been seeking to come up with a definition of cardinailty in that theory or variants, and I have been trying to produce models of that theory giving us some information about what can and can't be done. One possible sort of thing we can do in discussions like this is to produce proofs inside that base theory ZF - regularity. Whenever we produce such a proof, we are thereby showing that every model of the theory satisfies the sentece we just proved, So we have shown a property uniformly accross all models of the theory, and have shown that on this issue of deciding that one sentence all the models are similar to each other. Another sort of way to understand how a theory like this works is to argue model theoretically, and show some models of the theory satisfy a sentence. That allows the possibilty that some models satisfy that sentence and some do not. If so we don't have the uniformity. The models are not all the same in that regard. To see how one model works, we have to look deeper into it than merely knowing it satisifies the theory. In the other case of outright proofs from the theory, just knowing our model satisfied the theory was enough. So the model theoretic version has the potential to be more intricate and delicate than the base theory version. And indeed, if we are the case of a sentence working differently in different models, we are in the very case of incompleteness, when the base theory cannot do the job of supplying a proof. So if we are to still get useful information we need to go beyond that base theory, As we can do with model theory. We can see the model also from the outside, and have extra mathematical power beyond that base theory. So it is reasonable to expect to go outside the base theory to make progress, if indeed we are in the case of incompleteness. As would be the case if model building arguments really were working. So it a resonable exercise to step beyond the theory. It can happen that that way we get new information about the theory, which the results themselves show would have been impossible just from inside the theory. I will give a very simple example like this to illustrate my point. Later I will work my way back to our real point of discussion about singleton towers. But to make suggesitive analogies I will start with a really simple example, illustrating some of these points. So for the moment, replace the theory ZF - regularity by just the theory of = only, with no other non-logical axioms. This theory can express a sentence saying there is exactly one object in the universe. But it can't decide it. Suppose you want to rigourously show that can't be decided by the theory. One part of showing that could be to construct a two element model, thereby showing the theory doesn't prove that sentence. So to mathematically construct such a model, I could be working inside a larger theory that has 0 and 1 and proves they are not equal. Then I could construct my model as universe {0, 1}. How do I show the sentence is false in that model? I have to argue that 0 ~= 1. But any argument about properties of objects must use the definitions of these objects. So I must be using the definitions of 0 and 1, and my larger theory information about them, to conclude that. So I go outside of the theory of =, to get useful information about that theory. 0 amd 1 are not definable in the simple theory of equality. When you go back down to that language. you can't define them to distinguish them. Indeed the model admits an automorphism carrying one to the other. In that language the closest you can state relevant to this is just the ~= itself, which is not a satisfying proof. And the base theory can't prove that anyway, by the very incompleteness under discussion. So the final point from this simple example is that it can be a useful excercise to get information about a theory to view it from a more powerful theory, and use proofs in the more powerful language and theory to study models of the simpler theory. Let's step up a level, still simplified from ZF - regularity, but closer to it. Take the language and = and epsilon for membership, but lets consider the weaker theory of just usual extensionality. We can make a first model of this theory, from the badckground of conventional math, ie from outside this weak theory. Namely make the universe of the model be the usual integers Z, and make epsilon correspond so each n+1 is exactly the singleton of n. Ie: declare the membership relation E interoreting epilon has n E m <-> m = n+1 . This would be a model of one singleton tower, using that to correspond more closely to your writing than mine: infinite in both directions. Note that = in the model is interpreted as true =. The model satifies that object are = or not according to true =, ie as from the larger theory, not according to realations to E. I will illustrate this point with another structure, which will turn out not to be a model of extensionality. Let a, b, c be 3 distrinct objects, and make the model: a c \ / \ / \ / b where b E a b E c and nothing else. Then in this structure, we don't suddenly get a = c just because they have the same members, namely b. Instread still a ~= c. And what happens instead is extensionality fails. So this structure is not a model of the theory of extensionality. Let's return back to models of extensinanality. We previously had the Z model, with one singleton tower. Now let's consider a 2nd model, made from 2 disjoint isomprphic copies of Z, for example Z x 2 . And make each copy be like the previous Z singleton tower model, with no cross memberships between the 2 copies. So this is a model of extensionality with 2 distinct singleton towers. Why are the singleton towers distinct? For example why is <5, 0> ~= <5, 1> . Because those are ~= in the outer theory where we did the construction. And we are interpreting the = synbol in the constructed model by the true =, ie that of the outer working theory if you want to look at it that way. Yoy can't prove that from the simple theory of extesinionalty. Because it can't even define 5, 0, 1 or < , > . And there is nothing in that theory that tells you there must be 2 distinct disjoint singleton towers as this model has. Because the theorty of extensionality doesn't prove that: by the first model. Or indeed by a one element model with no singleton towers. Nothing from just the theory of extensionality seems to force the towers distinct. If you try to use extensionality to see <5, 0> ~= <5, 1> you seem to first need to show <4, 0> ~= <4, 1> to show they have different members which in turn makes you need <3, 0> ~= <3, 1> and so on to infinite regress. But that's not how we define the = interpretarion when we define a model. We define it by = in a larger theory, and there we have more power. So I can make this was a model of 2 disjoint singleton towers. But I can make another model with 3 towers. And if I want to make class sized models with definable classes, I can define a version of a proper class of copies of Z. These make models of extensionality, which is a smaller theory than ZF - regularity. 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 made models of ZF - regularity, based in the same core as above. In there I had various sets in the final model made from <alpha, n>. As n varies for alpha fixed it is a singleton tower. (A picky detail which may be a moment of confusion but not ultimately importannt: i took n > 0 only and towers only infinite downward not Z and two directions infinite as this time I copied from your discussion.) Why is the <alpha1, n1> tower disjoint from the <alpha2, n2> tower for alpha1 ~= alpha2 ? Because no tuples have <alpha1, n1> = <alpha2, n2> when aloha1 ~= alpha2. We don't argue these are distinct by invoking extensionality. Because that involves us in an infnite regress since the singleton descend infiintely. We use ~= from the background theory to get ~= in the constructed model. And knowing ~=, we get from that to extensionlity. Because those ~= pairs have ~= members: the singletons 1 step down. And those being ~= just depend on the background ~=, not more magic with extensionality. All this in analogy to {0, 1} as a two element model of the theory of equality. And we don't object to such a model by saying we are allowed to analyse it in the weak theory of equaltiy, and then complain that theory can't talk about 0 or 1, and can't prove anything ~=. I feel my previous articles, [1] and the others, already defined the models and outlined enough to show they have a proper class of singelons as you were asking. I have tried to see what could be pitfalls to interpretting those previous articles this way, and be explicit about the issues above. I am about to turn to your further writing. I will just note an important issue about that to come, is as I indicated above, it is possible to have disjoint singleton towers with no membership relations between them. So to return to your article: > First, a singleton tower, or what I call a "recursive singleton" can > be defined in the following manner: > > x is a singleton tower iff > x is singleton & > For all y ( y e TC(x) -> y is singleton ) & > For all y ( y e TC(x) -> ~ y e TC(y) ) > > So singleton towers are: hereditarily non circular singleton > hereditarily singletons. > > we can symbolize that as x={{{... ...}}} is a singleton tower. > > However for the sake of simplicity lets attach a natural number to > each > bracket in these singleton towers, i.e. let's number the brackets in > these towers, > > so let's say that x=0{1{2{... ...}}} > > i.e. the bracket number sequence of x is <0,1,2,3,.....> > so the outer bracket has the number 0, the one inside it has the > number 1, and the one inside it has the number 2, etc... > > Now we can see that the object y defined by the bracket sequence > <1,2,3,....> > will be inside x which is defined by the bracket sequence > <0,1,2,3,....> > > So in general the singleton tower xi+1 with bracket sequence of > <i+1,i+2,i+3,....> will be the member of the singleton tower xi with > bracket sequence of <i,i+1,i+2,i+3,...> > > I think this is clear, so we have xi+1 e xi for all singleton towers > defined above. > > Now lets take the tower were i=0, i.e. x0 with bracket sequence of > <0,1,2,3,....> > > Now what is the transitive closure of x0 > > This would be > > TC(x0)= {x1,x2,x3,..........} > > right! > > Now obviously TC(x0) is countable! Yes all ok. And as you say TC(x0) is countable. > Now let's perform iterative singleton operations on x0, and let's use > the negative integers for that purpose, so we'll have > > x(-1) = {x0} ,i.e. the bracket sequence of x(-1) is <-1,0,1,2,3,....> > x(-2) = { x(-1) } , i.e. the bracket sequence x(-2) is > <-2,-1,0,1,2,3,...> > . > . > . > (x(-i)) = {x(-i+1)} for all i=1,2,3,..... > > Now how many x(-i) we have? > > of course we have countably many of them! that is clear. Yes. And this would form one singleton tower, all parts iteratively connected, ordered like Z by membership in transtive closure. > Now the set of all singleton towers would be > > ST={...,x(-2),x(-1),x0,x1,x2,......} No. This numerical notation moves you around in one connected singleton tower. It doesn't let you move between disjoint towers unconnected by epsilon. As I showed in models, the simple ones above for just theory of extensionality, or [1] for ZF - regularity. > Which is countable! > > We cannot have more than those! > > If we take the singleton power of ST > denoted as P1(ST) (i.e. the set of all singleton subsets of ST) > , we'll only have ST itself > > ST=P1(ST) > > Now because of the following lemma in ZF minus Regularity: > > Lemma: For all x , for all y > > y e TC(x) if and only if there exist a finite sequence > <x0,x1,x2,...,xn> were x0 e x and > xi+1 e xi for every i=0,1,2,...,n-1 and y=xn. > > Then we cannot go down further, nor we can go up further, i.e. we > cannot have for example a singleton tower with a bracket sequence of > <0,1,2,.....,w,w+1,w+2,...> , this is forbidden by the lemma above, > also we cannot have a singleton tower with a bracket sequence of > <-w,.....,-2,-1,0,1,2,....>, or even <...,-2,-1,0,1,2,....>, all of > these cases are forbidden because of the lemma above. That lemma is ok for ome connected singleton tower. But it doensn't say what happens with other disjiont unconnected ones. > Also we cannot have a set with a bracket sequence of for example > <0,2,3,4,...>, because this would be equal to the x1 i.e. the > singleton > tower with bracket sequence of <1,2,3,....>, also we cannot > have sets with bracket sequences of for example > <1,3,5,7,...>, or <1,2,3,5,7,11,...>, these have gaps in their > sequences > which is not compatible with the definition of singleton powers. The notatin is not well defined accross such independent singleton towers. > Actually I don't see how we can ever have an uncountable number of > these singleton towers?! You can't derive it just from ZF - regularity. But you can constuct it as [1] into come models. > Actually even if we suppose that we can have these gap sequences > above, and by them one might conclude that we can have > an uncountable number of these singleton towers, but still the issue > remains, > how can we have a proper class of them??? By putting a proper class of indices into the definition from outside in [1] . > Because of the lemma above it is clear that Singleton powering of ST > will stop giving different sets at some point, i.e. the process is > exhaustive! > So the class of all singleton towers, which is the cardinal number one > would be a set! Each singleton tower only has countably many elements, as you wrote. But there can be many of them. even a proper class as [1]. And they don't generate from each other by such operations. > I might be mistaken though, but I would like to know how can we have > an uncountable number of these singleton towers? and even more > how can we have a proper class of them? I want a proof from within > the model, and not from outside it. > > > Zuhair I have to argue from outside. But that's ok. It still shows me there is a model. From the outside theory which we believe. How do you show the theory iof equality has 2 element models, if you are only allowed to argue from inside the pure theory of equality? -- David Libert ah170(a)FreeNet.Carleton.CA
From: zuhair on 27 Dec 2009 20:09 On Dec 26, 2:33 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > zuhair (zaljo...(a)gmail.com) writes: > > I have to argue from outside. But that's ok. It still shows me there > is a model. From the outside theory which we believe. > > How do you show the theory iof equality has 2 element models, if > you are only allowed to argue from inside the pure theory of > equality? > > -- > David Libert ah...(a)FreeNet.Carleton.CA First I want to thank you a lot a lot really, for writing such a nice account, that is simple, clear, and straightforward. I appreciate the time you've spent with all the subjects that I have raised, and I am very thankful really. Actually I was suspecting the possibility of singleton towers with disjoint transitive closures, and I was suspecting that you were using that. 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. But still I do think that even if you have these transitively disjoint singleton towers, still you cannot have a proper class of them! The reason is the following, Let me explain the bracket sequence methodology that I began with because I see it easier. I noticed that you misunderstood them actually. x_i is the singleton tower with bracket sequence of <i,i+1,i+2,.....> so suppose that i=0 then x_0 is the *singleton tower* with bracket sequence of <0,1,2,3,....> you can represent that as x_0 = 0{1{2{... ...}}} so the outer bracket has the number 0, the one inside has number 1, the one inside has number 2, and so one. to understand this notation notice that the singleton tower x_1 would have the bracket sequence of <1,2,3,....> and you will have x_1 e x_0 so x_1 is a *singleton tower* and it is the sole member of the singleton tower x_0 so each x_i is a singleton tower Now each x_i+1 is a singleton tower and it is a member of the singleton tower x_i. I think this is pretty clear. Now the transitive closure of x_0 is NOT a singleton tower as you thought, the transitive closure of x_0 is actually not singleton at all, it has countably many singleton towers in it (review the definition of singleton tower as a *singleton* that is hereditarily singleton and also that is non circular). So TC(x_0) is NOT a singleton tower, it is a set of singleton towers yes, but since it is itself not singleton, that's why it is not a singleton tower itself. TC(x_0) = {x_1, x_2, x_3,.....} Notice that Each x_i for i=1,2,3,.... is a singleton tower. Of course all sets in TC(x_0) are as you said not disjoint at their transitive closures, i.e. we have the following For all x,y e TC(x_0) ( x e TC(y) or y e TC(x) ). That is definitely true. right. And even my Omega powering of TC(x_0) would at the end yield what you said "singleton towers" that are not disjoint at their transitive closures. 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. Actually I came to realize lately that we might not be able to have x-recursive sets were x>1. or even if we have these, then this must come to an end, and I think we cannot have x-recursive sets were x>= power(omega). All of these results make me think that we can actually define Cardinality using Scott trick. Suppose every x such that x-recursive cardinal exist entail that x < power(omega). Then the class of *all* x-recursive cardinals is a set! lets call it RC Now RC will be the base set, and we build a hierarchy similar to the Cumulative hierarchy ( a permutation model ) were RC is the base set. so V0=RC for every successor ordinal i , Vi+1= Power(Vi) for every limit ordinal j , Vj = Union(i) Vi were i e j. Obviously cardinality can be defined using Scott trick. So it seems that my latest version of strong Extensionality that shuns circular sets from existence, would actually work to save defined cardinality. Zuhair
From: zuhair on 28 Dec 2009 07:17 On Dec 27, 8:09 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Dec 26, 2:33 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > > > zuhair (zaljo...(a)gmail.com) writes: > > > I have to argue from outside. But that's ok. It still shows me there > > is a model. From the outside theory which we believe. > > > How do you show the theory iof equality has 2 element models, if > > you are only allowed to argue from inside the pure theory of > > equality? > > > -- > > David Libert ah...(a)FreeNet.Carleton.CA > > First I want to thank you a lot a lot really, for writing such a nice > account, that is simple, clear, and straightforward. I appreciate the > time you've spent with all the subjects that I have raised, and I am > very thankful really. > > Actually I was suspecting the possibility of singleton towers with > disjoint transitive closures, and I was suspecting that you were using > that. > > 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. > > But still I do think that even if you have these transitively disjoint > singleton towers, still you cannot have a proper class of them! > > The reason is the following, Let me explain the bracket sequence > methodology that I began with because I see it easier. > > I noticed that you misunderstood them actually. > > x_i is the singleton tower with bracket sequence of <i,i+1,i+2,.....> > > so suppose that i=0 then x_0 is the *singleton tower* with > bracket sequence of <0,1,2,3,....> > > you can represent that as x_0 = 0{1{2{... ...}}} Actually this notation is confusing because one might think that 1,2,.. are members of x_0, but they are not really, they are only markers for the brackets themselves, may be a better notation would be: x_0 = 0_{ 1_{ 2_{... ...}}} > > so the outer bracket has the number 0, the one inside has number 1, > the one inside has number 2, and so one. > > to understand this notation notice that the singleton tower x_1 would > have the bracket sequence of <1,2,3,....> > > and you will have x_1 e x_0 > > so x_1 is a *singleton tower* and it is the sole member of the > singleton tower x_0 > > so each x_i is a singleton tower > > Now each x_i+1 is a singleton tower and it is a member of the > singleton tower x_i. > > I think this is pretty clear. > > Now the transitive closure of x_0 is NOT a singleton tower as you > thought, the transitive closure of x_0 is actually not singleton at > all, it has countably many singleton towers in it (review the > definition of singleton tower as a *singleton* that is hereditarily > singleton and also that is non circular). > > So TC(x_0) is NOT a singleton tower, it is a set of singleton towers > yes, but since it is itself not singleton, that's why it is not a > singleton tower itself. > > TC(x_0) = {x_1, x_2, x_3,.....} > > Notice that Each x_i for i=1,2,3,.... is a singleton tower. > > Of course all sets in TC(x_0) are as you said not disjoint at their > transitive closures, i.e. we have the following > > For all x,y e TC(x_0) ( x e TC(y) or y e TC(x) ). > > That is definitely true. right. > > And even my Omega powering of TC(x_0) would at the end yield what you > said > "singleton towers" that are not disjoint at their transitive closures. > > 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. > > Actually I came to realize lately that we might not be able to have > x-recursive sets were x>1. or even if we have these, then this must > come to an end, > and I think we cannot have x-recursive sets were x>= power(omega). > > All of these results make me think that we can actually define > Cardinality using Scott trick. > > Suppose every x such that x-recursive cardinal exist entail that x < > power(omega). > > Then the class of *all* x-recursive cardinals is a set! lets call it > RC > > Now RC will be the base set, and we build a hierarchy similar to the > Cumulative hierarchy ( a permutation model ) were RC is the base set. > > so V0=RC > for every successor ordinal i , Vi+1= Power(Vi) > for every limit ordinal j , Vj = Union(i) Vi were i e j. > > Obviously cardinality can be defined using Scott trick. > > So it seems that my latest version of strong Extensionality that > shuns circular sets from existence, would actually work to save > defined cardinality. > > Zuhair
From: zuhair on 28 Dec 2009 07:24 On Dec 27, 8:09 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Dec 26, 2:33 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > > > zuhair (zaljo...(a)gmail.com) writes: > > > I have to argue from outside. But that's ok. It still shows me there > > is a model. From the outside theory which we believe. > > > How do you show the theory iof equality has 2 element models, if > > you are only allowed to argue from inside the pure theory of > > equality? > > > -- > > David Libert ah...(a)FreeNet.Carleton.CA > > First I want to thank you a lot a lot really, for writing such a nice > account, that is simple, clear, and straightforward. I appreciate the > time you've spent with all the subjects that I have raised, and I am > very thankful really. > > Actually I was suspecting the possibility of singleton towers with > disjoint transitive closures, and I was suspecting that you were using > that. > > 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. > > But still I do think that even if you have these transitively disjoint > singleton towers, still you cannot have a proper class of them! > > The reason is the following, Let me explain the bracket sequence > methodology that I began with because I see it easier. > > I noticed that you misunderstood them actually. > > x_i is the singleton tower with bracket sequence of <i,i+1,i+2,.....> > > so suppose that i=0 then x_0 is the *singleton tower* with > bracket sequence of <0,1,2,3,....> > > you can represent that as x_0 = 0{1{2{... ...}}} > > so the outer bracket has the number 0, the one inside has number 1, > the one inside has number 2, and so one. > > to understand this notation notice that the singleton tower x_1 would > have the bracket sequence of <1,2,3,....> > > and you will have x_1 e x_0 > > so x_1 is a *singleton tower* and it is the sole member of the > singleton tower x_0 > > so each x_i is a singleton tower > > Now each x_i+1 is a singleton tower and it is a member of the > singleton tower x_i. > > I think this is pretty clear. > > Now the transitive closure of x_0 is NOT a singleton tower as you > thought, the transitive closure of x_0 is actually not singleton at > all, it has countably many singleton towers in it (review the > definition of singleton tower as a *singleton* that is hereditarily > singleton and also that is non circular). > > So TC(x_0) is NOT a singleton tower, it is a set of singleton towers > yes, but since it is itself not singleton, that's why it is not a > singleton tower itself. > > TC(x_0) = {x_1, x_2, x_3,.....} > > Notice that Each x_i for i=1,2,3,.... is a singleton tower. > > Of course all sets in TC(x_0) are as you said not disjoint at their > transitive closures, i.e. we have the following > > For all x,y e TC(x_0) ( x e TC(y) or y e TC(x) ). > > That is definitely true. right. > > And even my Omega powering of TC(x_0) would at the end yield what you > said > "singleton towers" that are not disjoint at their transitive closures. > > 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. > > Actually I came to realize lately that we might not be able to have > x-recursive sets were x>1. or even if we have these, then this must > come to an end, > and I think we cannot have x-recursive sets were x>= power(omega). > > All of these results make me think that we can actually define > Cardinality using Scott trick. > > Suppose every x such that x-recursive cardinal exist entail that x < > power(omega). > > Then the class of *all* x-recursive cardinals is a set! lets call it > RC Sorry: the set union of the set of all recursive cardinals, would be the base set, so it would be RC. > > Now RC will be the base set, and we build a hierarchy similar to the > Cumulative hierarchy ( a permutation model ) were RC is the base set. > > so V0=RC > for every successor ordinal i , Vi+1= Power(Vi) > for every limit ordinal j , Vj = Union(i) Vi were i e j. > > Obviously cardinality can be defined using Scott trick. > > So it seems that my latest version of strong Extensionality that > shuns circular sets from existence, would actually work to save > defined cardinality. > > Zuhair
From: David Hartley on 28 Dec 2009 07:22
In message <88909956-acd2-407d-a673-c42065e5a093(a)k19g2000yqc.googlegroups.com>, zuhair <zaljohar(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? -- David Hartley |