Uncountability of the Rationals?
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From: Tony Orlow on 8 Jun 2010 08:45 On Jun 8, 1:08 am, Virgil <Vir...(a)home.esc> wrote: > In article > <d6c03228-ad6c-4d32-833a-b742f3b09...(a)y4g2000yqy.googlegroups.com>, > Tony Orlow <t...(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"? In that greater quantities occur after smaller quantities. Monotonically increasing. You've heard of it. > > 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? > I've done this about a million times. Did you read my post to Transfer explaining the T-riffics? > > > > > 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. > > To you, perhaps. > > > > > 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. When you have to snip the end of my sentence in order to make my statement sound stupid the you're stretching and making yourself look bad. > \ > > > > > > > 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. Only in the minds of the Garden Guard Dogs. > > > 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. With infinite case induction, if x>2 then the above is true for x, period. > > > > > > 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. Then you haven't paid any attention to it. > > > 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.- Hide quoted text - I'll bring your urn. > > - Show quoted text -- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - :) Tony
From: Tony Orlow on 8 Jun 2010 08:47 On Jun 8, 1:37 am, David R Tribble <da...(a)tribble.com> wrote: > 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- Hide quoted text - > > - Show quoted text - By the axiom of extensionality, as sets they are identical. As sequences they differ. If a sequence contains the same members as another which is monotonically increasing, then that other can be used to calculate the bigulosity of both sets/sequences. Tony
From: Tony Orlow on 8 Jun 2010 08:52 On Jun 8, 2:00 am, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-08, Tony Orlow <t...(a)lightlink.com> wrote: > > > I don't even know the basics? > > Correct. > > > Excuse me, but there are several unanswered questions, even in your > > provincial neighborhood. There is the continuum hypothesis > > What in particular do you think is unanswered about the continuum > hypothesis? Whether aleph_1 is equal to c, or whether there exists an aleph between aleph_0 and c. > > > there is a non-existent but "provably extant" well ordering of the > > reals > > What is nonexistent about a well ordering of the reals? Ummm... none has ever be explicitly discovered or defined. > > > Remember my big challenge? What is the width of a countably infinite > > complete list of digital numbers? > > In what salient way does your "big challenge" differ from "what is the > width of a snarflek"? Do you think such a comment makes ME look bad? Do you not understand English? > > > This is the same question as CH. > > Only in your own little logically inconsistent corner of mathematical > theory. > > > That's all solved in my quadrant of the cosmos. > > Well of course it's solved in your quadrant - every inconsistent > theory has proofs for all propositions. > > - Tim Sorry, Tim, but I'm unlikely to respond to any more of your comments, since they are simply vacuous spewage. Good luck with that. Tony
From: Tony Orlow on 8 Jun 2010 08:55 On Jun 8, 4:44 am, Tim Little <t...(a)little-possums.net> wrote: <snip> > > > 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 > > You say that x^2 < 2^x. Try x = 3 or 4. My bad. That is true for x>4. Oops. Is omega greater than 4? > > > You know IFR makes vastly more distinctions than cardinality. You > > know infinite-case induction is currently workable, > > I know no such things. > > - Tim Tony
From: MoeBlee on 8 Jun 2010 09:25
On Jun 7, 7:54 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 7, 5:23 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 5, 12:44 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > On Jun 4, 4:09 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > On Jun 4, 2:50 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > (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. No, you've not shown a general definition of 'usual ordering'. > The "usual" ordering of a quantitative set is ... > quantitative. That's what you mean by usual - ordered along "the > line". Would you please LISTEN? Not every set comes equipped with a "line". There are all kinds of sets other than the naturals, rationals, reals, and a few others that have a "canonical" line associated with them. If you have a definition of 'the usual ordering of X' for arbitrary sets X then let's hear it. > > (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. I never denied that you've posted verbiage about "infinite-case induction" and other manner of things. > > 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. "mapping formula". The private terminology of Tony Orlow, unmoored from any theory or definitive glossary of terms reducing to primitives. Look, two sets have a bijection or not, no matter what orderings there are on the sets. > > 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. > Pretend otherwise if you so desire, but don't pretend I won't remember > how things are distorted. I'm used to that. I live among highfalootin' > apes. Yes, your part in the conversation now has become a lot of jumping up and down and beating of chest. > > 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. IFR is without flaw, and satisfies intuitive > notions without problem until people start using complex mapping > formulae which require the axiom of extension to resolve. Yes, "IFR", "complex mapping formulae", "axiom of extension", "resolve". > AppendixSpleen Indeed, you're tending to the splenetic now as mere appendage to any substantive conversation. MoeBlee |