Uncountability of the Rationals?
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From: Virgil on 7 Jun 2010 23:00 In article <6ee0b488-6ea9-41b5-8779-bf1309b3c85f(a)k39g2000yqd.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 7, 5:23�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > Okay, but you miss two points: > > > > (1) For only occasional sets do we have a "usual" ordering. In > > general, there is no definition provided (especially by you) for "the > > usual ordering of a set". So your analysis in that respect doesn't > > even get off the ground. > > Disingenuous. The "usual" ordering of a quantitative set is ... > quantitative. That's what you mean by usual - ordered along "the > line". How does a "quantitative set" differ from other sets? Is it a special case of a linearly ordered , or totally ordered set? > > > > > (2) You have at least suggested below that bijection is a superficial > > view of the notion of size. But we find that your notion of size based > > on ordering is even more superficial since it doesn't penetrate the > > matter any further than a grammar school student's notion of ordering. > > Incorrect. I have made clear the infinite-case induction argument, and > how it is applied to formulae applied to cases greater than the > finite, in general. That "infinite-case induction argument" may appear clear to you, but your expostion of it is not clear to anyone else. > > > When we look at orderings on sets in a more full way, we find that > > sets can be ordered in many ways (and as mentioned above, there is no > > general definition of "the usual ordering of a set") so that a notion > > of 'size' should accomodate all orderings, and indeed bijection does > > cut across all orderings. > > That's superficial, without regard to salient details, such as the > mapping formula. Anything that TO is at able to comprehend must be superficial, and injection and bijection of all but trivially small sets are not dependent on any particular mapping formula. > > > > > > > > with a countably infinite number of > > > > > rationals lying quantitatively between any two given naturals, can > > > > > you? Do you not see that in some sense there appear to be more > > > > > rationals than naturals? Not as merely sets, though as ORDERED sets one appears to be "denser" than the other. > > > > > > Yes, but some investigation shows that that is a superficial view, > > > > since we see also that we can order the naturals densely and order the > > > > rationals discreetly. > > > > > Perhaps simple bijection as a proof of equinumerosity is superficial. > > > That's also a possibility. :) > > > > First, it's definition. Second, so what that it's possible that the > > method bijection may be superficial? You've not shown that it is > > superficial, while I have shown how your view IS superficial, while > > also bijection conforms to a basic intuition carried from the finite. > > It's simplistic, and I have demonstrated in many ways how it is so. If you mean is it simple, I concur, but "simplistic" is undeservedly pejorative. > Pretend otherwise if you so desire, but don't pretend I won't remember > how things are distorted. As a matter of fact, you often contribute significantly a good deal of distortion to anything you comment on.. > I'm used to that. I live among highfalootin' apes. Your family? > > > If there were a place to at least START thinking what size would be in > > the infinite it is reasonable to at least countenance that it be as in > > the finite, based on bijection. Then that presumption may be > > overturned IF there were a better notion with which to do the > > overturning. But you've not shown any better notion. That YOU > > personally think that your own notions (still none of them put into > > theory form) are superior is not a convincing basis. > > You know that's not true. We do not know any such thing!
From: Virgil on 7 Jun 2010 23:06 In article <53b4dfa9-dae3-4760-bbee-212f7b12bea4(a)j4g2000yqh.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 7, 5:27�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 5, 12:50�am, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > > > > > > > On Jun 4, 4:24�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > > On Jun 4, 3:20�pm, David R Tribble <da...(a)tribble.com> wrote: > > > > > > > Tony Orlow wrote: > > > > > > One might think there were something like aleph_0^2 rationals, but > > > > > > that's not standard theory. > > > > > > > Actually, there are Aleph_0^2 rationals. And Aleph_0^2 = Aleph_0. > > > > > > Orlow can't be bothered to learn such basics. > > > > > > MoeBlee > > > > > Piffle to you both. I already stated that very fact very early in this > > > thread. > > > > State it all you want; your disclaimer doesn't then just push the > > matter out of the way. In fact, it is a salient matter as to the > > nature and quality of conversation with you. That is, you just keep > > spouting on a subject of which you don't even know the basics. You're > > welcome to take how many years you wish to formulate whatever theory; > > and quite welcome to critique set theory in the meantime; but what you > > do is spout your critique from a platform of ignorance and > > misconception on the subject, thus making much of your postings piles > > of misinformation and misconception that need to be cleaned up. > > > > MoeBlee- Hide quoted text - > > > > - Show quoted text - > > I don't even know the basics? Excuse me, but there are several > unanswered questions, even in your provincial neighborhood. There is > the continuum hypothesis, there is a non-existent but "provably > extant" well ordering of the reals. Remember my big challenge? What > is the width of a countably infinite complete list of digital numbers? > This is the same question as CH. That's all solved in my quadrant of > the cosmos. It's computable. That should be rather satisfying, and not > at all frustrating, given that your motives are pure. I do not see TO having either a proof of the continuum hypothesis or a proof of its negation. I do not see TO having either an explicit well-ordering of the reals or any proof that none can exist. So those issues have not been "solved" nor "computed" in TO's quadrant of the cosmos, as he claims. And I have absolutely no faith ever in the purity of TO's motives.
From: Virgil on 7 Jun 2010 23:08 In article <5483761b-b07b-4dd9-8422-98d5dcbd9451(a)g19g2000yqc.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 7, 5:29�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 5, 1:06�pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > or is less dense in the natural quantitative order > > > > There is no general definition that we have (let alone that you have > > provided) of "the natural quantitative order". > > > > MoeBlee > > It's simply a matter of the axioms of order. Examine them alone for a > bit. The order axioms nowhere mention "the natural quantitative order" at all. So that TO is again blowing hot and cold in the same breath.
From: Virgil on 7 Jun 2010 23:10 > Moe, you have a point. I have from the beginning objected to the > status quo in this regard because it has always offended my > sensibilities. And there are very few whose sensibilities more deserve being offended than TO. Possibly AP.
From: Tony Orlow on 8 Jun 2010 00:38
On Jun 7, 11:00 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <6ee0b488-6ea9-41b5-8779-bf1309b3c...(a)k39g2000yqd.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > On Jun 7, 5:23 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > Okay, but you miss two points: > > > > (1) For only occasional sets do we have a "usual" ordering. In > > > general, there is no definition provided (especially by you) for "the > > > usual ordering of a set". So your analysis in that respect doesn't > > > even get off the ground. > > > Disingenuous. The "usual" ordering of a quantitative set is ... > > quantitative. That's what you mean by usual - ordered along "the > > line". > > How does a "quantitative set" differ from other sets? Is it a special > case of a linearly ordered , or totally ordered set? > As always, O Virgil, I respect that you finally end up asking real questions, after some witty repartee :) A quantitatively ordered set is one wherein each member is either greater than, less than, or equal to every other member. Yes/no? Trichotomy applies? If neither x<y or x>y then x=y, noyes? > > > > > (2) You have at least suggested below that bijection is a superficial > > > view of the notion of size. But we find that your notion of size based > > > on ordering is even more superficial since it doesn't penetrate the > > > matter any further than a grammar school student's notion of ordering.. > > > Incorrect. I have made clear the infinite-case induction argument, and > > how it is applied to formulae applied to cases greater than the > > finite, in general. > > That "infinite-case induction argument" may appear clear to you, but > your expostion of it is not clear to anyone else. I think it is up to others to speak for themselves, as it is up to you to do so. If it is not clear to you, then please ask a clarifying question. Under the criterion of difference in the infinite case as a basis for a formulaic inequality, the difference that either remains or grows (more usually) stands as a basis for application of the '<' symbol. Extensions should be as simple as possible, and still make as many distinctions as possible. That's logical beauty. > > > > > > When we look at orderings on sets in a more full way, we find that > > > sets can be ordered in many ways (and as mentioned above, there is no > > > general definition of "the usual ordering of a set") so that a notion > > > of 'size' should accomodate all orderings, and indeed bijection does > > > cut across all orderings. > > > That's superficial, without regard to salient details, such as the > > mapping formula. > > Anything that TO is at able to comprehend must be superficial, and > injection and bijection of all but trivially small sets are not > dependent on any particular mapping formula. > Dear, they are. We've discussed IFR. You know it already. But, I appreciate your efforts. Thanx. > > > > > > > > with a countably infinite number of > > > > > > rationals lying quantitatively between any two given naturals, can > > > > > > you? Do you not see that in some sense there appear to be more > > > > > > rationals than naturals? > > Not as merely sets, though as ORDERED sets one appears to be "denser" > than the other. > Yes, there is no explicitly defined infinite set which is not also a sequence, or some recursive structure with even more than one dimension (child per level). Quantitatively, some sets are denser according to the "usual" ordering (along the line). At least that's what I was just recently told.... > > > > > > > Yes, but some investigation shows that that is a superficial view, > > > > > since we see also that we can order the naturals densely and order the > > > > > rationals discreetly. > > > > > Perhaps simple bijection as a proof of equinumerosity is superficial. > > > > That's also a possibility. :) > > > > First, it's definition. Second, so what that it's possible that the > > > method bijection may be superficial? You've not shown that it is > > > superficial, while I have shown how your view IS superficial, while > > > also bijection conforms to a basic intuition carried from the finite. > > > It's simplistic, and I have demonstrated in many ways how it is so. > > If you mean is it simple, I concur, but "simplistic" is undeservedly > pejorative. I don't mean to be demeaning, except to the hopelessly unreceptive and aggressively rejective. I agree that it was worthwhile for Georg to resurrect the comparison of infinities and try to discern between them as much as possible. I named him with Dedekind as founders along the way, and Cantor paid dearly for his devotion to this subject, suffering rejection and derision along the way, and spending much of his later life under supervision. It's sad. However Aristotle simply made a distinction between the two, and doubted the existence of the latter, Galileo considered bijection and wrote it off as a measure for infinite sequences, and Cantor picked up the ball. Maybe it wasn't the right time. Wait - didn't they all end up under supervision? In any case, you might want to give me a reason why you think infinite- case induction does not work for espressions of inequality on formulas of x based on differences that do not have a limit of 0 as x increases without bound. Usually they increase without bound. For instance, for x>2: x+2<x*2<x^2<2^x<x^x Um, true or false? omega>2? > > > Pretend otherwise if you so desire, but don't pretend I won't remember > > how things are distorted. > > As a matter of fact, you often contribute significantly a good deal of > distortion to anything you comment on.. How so? Let's weigh in. > > > I'm used to that. I live among highfalootin' apes. > > Your family? > Yep, we faloot high amongst these here trees. Havabanana. :) If you don't admit to being an ape, I challenge you to make a distinction that's "rigorous". > > > > If there were a place to at least START thinking what size would be in > > > the infinite it is reasonable to at least countenance that it be as in > > > the finite, based on bijection. Then that presumption may be > > > overturned IF there were a better notion with which to do the > > > overturning. But you've not shown any better notion. That YOU > > > personally think that your own notions (still none of them put into > > > theory form) are superior is not a convincing basis. > > > You know that's not true. > > We do not know any such thing! You do too. You know IFR makes vastly more distinctions than cardinality. You know infinite-case induction is currently workable, and how N=S^L works with IFR. Virgil, you know it all. You know you're invited as a guest when I accept the Fields Medal... :) Tony |