Uncountability of the Rationals?
in [Math]
|
Prev: Collatz conjecture
Next: Beginner-ish question
From: Tony Orlow on 8 Jun 2010 00:43 On Jun 7, 11:06 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <53b4dfa9-dae3-4760-bbee-212f7b12b...(a)j4g2000yqh.googlegroups.com>, > Tony Orlow <t...(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. Not a bigulous issue in the least, but tht's okay... > > I do not see TO having either an explicit well-ordering of the reals or > any proof that none can exist. Nah, I gave up on that "well" ordering thing. It was making me sick. But, anyway... > > So those issues have not been "solved" nor "computed" in TO's quadrant > of the cosmos, as he claims. One completely resolved, one dismissed, sweetie. Don't worry. > > And I have absolutely no faith ever in the purity of TO's motives.- Hide quoted text - > > - Show quoted text - I would never do you wrong, babe. BTW, you look really cute in that dress. Motives? Whaddya mean? I don't have to take this abuse... Audi5000 T:)ny
From: David R Tribble on 8 Jun 2010 00:52 MoeBlee wrote: >> (1) For only occasional sets do we have a "usual" ordering. In >> general, there is no definition provided (especially by you [Tony]) for "the >> usual ordering of a set". So your analysis in that respect doesn't >> even get off the ground. > David R Tribble wrote: >> Indeed. Here are a few ways to write the set of naturals, >> all of them being exactly the same set: >> { 0, 1, 2, 3, 4, 5, 6, ... } >> { 0, 2, 1, 4, 3, 6, 5, ... } >> { 0, 2, 4, 6, ..., 1, 3, 5, 7, ... } >> { 0, 1, 3, 4, 6, 7, 9, 10, ..., 2, 5, 8, 11, ... } >> { ..., 5, 4, 3, 2, 1, 0 } >> { 0, ..., 4, 3, 2, 1 } >> >> Tony must realize that a set is simply a collection, >> specifically an *unordered* collection. Just because all >> of its members are natural numbers (or reals, or rationals, >> or whatever) does not automatically imbue it with an >> ordering. > Tony Orlow wrote: > Why are all of these infinite sets you cite some kind of sequence? They're not. Not a single one. And (I repeat), they are all exactly the same set. Again, you come across as though you truly don't get it, intentionally, or sarcastically, or otherwise. -drt
From: Tony Orlow on 8 Jun 2010 01:00 On Jun 8, 12:52 am, David R Tribble <da...(a)tribble.com> wrote: > MoeBlee wrote: > >> (1) For only occasional sets do we have a "usual" ordering. In > >> general, there is no definition provided (especially by you [Tony]) for "the > >> usual ordering of a set". So your analysis in that respect doesn't > >> even get off the ground. > > David R Tribble wrote: > >> Indeed. Here are a few ways to write the set of naturals, > >> all of them being exactly the same set: > >> { 0, 1, 2, 3, 4, 5, 6, ... } > >> { 0, 2, 1, 4, 3, 6, 5, ... } > >> { 0, 2, 4, 6, ..., 1, 3, 5, 7, ... } > >> { 0, 1, 3, 4, 6, 7, 9, 10, ..., 2, 5, 8, 11, ... } > >> { ..., 5, 4, 3, 2, 1, 0 } > >> { 0, ..., 4, 3, 2, 1 } > > >> Tony must realize that a set is simply a collection, > >> specifically an *unordered* collection. Just because all > >> of its members are natural numbers (or reals, or rationals, > >> or whatever) does not automatically imbue it with an > >> ordering. > > Tony Orlow wrote: > > Why are all of these infinite sets you cite some kind of sequence? > > They're not. Not a single one. > > And (I repeat), they are all exactly the same set. > > Again, you come across as though you truly don't get it, > intentionally, or sarcastically, or otherwise. > > -drt- Hide quoted text - > > - Show quoted text - Wow. Those aren't sequences, with or without internal limit "ordinals"? Look again. What is your personal definition of a sequence, again? Oy, Tony
From: Virgil on 8 Jun 2010 01:08 In article <d6c03228-ad6c-4d32-833a-b742f3b09646(a)y4g2000yqy.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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? Then how, if at all, do your "quantitatively ordered sets" differ from the more standard "ordered sets" or "totally ordered sets" or "linearly ordered sets"? Unless there is some difference, I see no reason for creating a new name for what is already satisfactorily and multiply named. > > > > > > > > > (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. My questions: What do you mean by "infinite-case induction"? Is there some more standard term for it, as there is for your "quantitatively ordered sets" or it it an invention of your own. Can you give an example of it? > > > > 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. Not at all. For all but finite sets there are for every instance of a surjection mapping formula from a given set an infinity of alternates each of which is just as satisfactory. > 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 The rationals standardly ordered are not a sequence. The reals standardly ordered are not a sequence. The complexes, as standardly constructed, are not even ordered, much less a sequence. \ > > > > > 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. Much of what you propose deserves to be aggressively rejected. > 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, As I have never been able to make sense of what you vaguely denote as infinite-case induction. > > x+2<x*2<x^2<2^x<x^x Which is senseless unless x is restricted to a specific domain. > > > > 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. I do not know that what you call IFR makes any sense at all. > You know infinite-case induction is currently workable No I don't. I am much to picky about what I "know" to know that. > and how N=S^L works with IFR. No I don't. > Virgil, you know it all. You know you're > invited as a guest when I accept the Fields Medal... If I really know it all then what I know of your IFR makes it nonsense. Besides, re your invitation, I cannot possibly live that long.
From: David R Tribble on 8 Jun 2010 01:37
MoeBlee wote: >> (1) For only occasional sets do we have a "usual" ordering. In >> general, there is no definition provided (especially by you [Tony]) for "the >> usual ordering of a set". So your analysis in that respect doesn't >> even get off the ground. > David R Tribble wrote: >> Indeed. Here are a few ways to write the set of naturals, >> all of them being exactly the same set: >> { 0, 1, 2, 3, 4, 5, 6, ... } >> { 0, 2, 1, 4, 3, 6, 5, ... } >> { 0, 2, 4, 6, ..., 1, 3, 5, 7, ... } >> { 0, 1, 3, 4, 6, 7, 9, 10, ..., 2, 5, 8, 11, ... } >> { ..., 5, 4, 3, 2, 1, 0 } >> { 0, ..., 4, 3, 2, 1 } >> >> Tony must realize that a set is simply a collection, >> specifically an *unordered* collection. Just because all >> of its members are natural numbers (or reals, or rationals, >> or whatever) does not automatically imbue it with an >> ordering. > Tony Orlow wrote: >> Why are all of these infinite sets you cite some kind of sequence? > David R Tribble wrote: >> They're not. Not a single one. >> And (I repeat), they are all exactly the same set. >> >> Again, you come across as though you truly don't get it, >> intentionally, or sarcastically, or otherwise. > David R Tribble wrote: > Wow. Those aren't sequences, with or without internal limit > "ordinals"? Look again. Nope, sorry, I don't see any sequences, just a list of sets. Actually a list of the *same* set written in a few different ways. You're reading more into "set" than is there. Again, you're making yourself look foolish, acting as though you really don't understand what a "set" is. > What is your personal definition of a sequence, again? Fairly close to the actual definition. Essentially, it's a set that maps the naturals to a function (a function being a set itself, of course, within set theory). Here's a few example sequences: S1 = 0, 1, 2, 3, 4, 5, 6, ... S2 = 0, 2, 1, 4, 3, 6, 5, ... S3 = 0, 2, 4, 6, ..., 1, 3, 5, 7, ... S4 = 0, 1, 3, 4, 6, 7, 9, 10, ..., 2, 5, 8, 11, ... You'll notice that all of these particular sequences are *different* (which is why I gave them different names), even though they all contain the same values (all of the naturals, as it happens, in these examples). -drt |