Uncountability of the Rationals?
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From: David R Tribble on 7 Jun 2010 16:41 David R Tribble wrote: >> Tony's ideas [...] . I.e., the sequence of >> naturals (eventually) contains infinite naturals, infinite sets >> must contain at least one infinite member, all numbers >> must have digital representations, and a few other wacky >> ideas. > Tony Orlow wrote: > Oh, David, don't even go there. > I have never claimed the standard naturals include infinite > values, nor denied that the set of points in [0,1] is infinite in size > but contains no infinite values, nor claimed that there exists any > finite digital representation of any transcendental number nor a last > digit thereof, or any such wacky ideas. Your disingenuity [sic] is most > unbecoming, and I would appreciate it if you not spin fabrications > about what you believe to be my dysfunction. The splinter you perceive > in my eye is but a reflection of the log in yours. Here is some more to deny. I found this thread from June 2005: http://groups.google.com/group/sci.math/msg/9ab9be0a33ae6119 In there you say things like: | Then you do not have an infinite set, but an "unbounded" set. | The standard set N has maximum member N, but that | doesn't mean that one can't count higher. Think of N | as 999...999, and N+1 as 000...001:000...000 | You still haven't told me what I did wrong in my S^L proof. | How do you get infinite numbers of strings without having | infinitely long strings? That last quote sounds a lot like what you claim you never said about infinite members of infinite sets. -drt
From: Tony Orlow on 7 Jun 2010 16:57 On Jun 7, 3:59 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <9b865fc2-1836-4aa3-9f5f-016f04300...(a)u26g2000yqu.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > The argument that the truly infinite set of naturals must include an > > infinite natural has nothing particular to do with N=S^L. That's a > > simple matter of the nth element of any initial segment of N being > > equal to n and existing for any segment of size n or greater. > > One can have initial segments of infinite well ordered sets which have a > last member and initial segments which do NOT have a last member, and > the two types are quite distinct. TO attempts to overlook the difference > and conflate the two types. > > In particular, for infinite well-ordered sets there is no necessity for > their "initial sets" all to have largest members, and some of those > initial sets necessarily don't have largest members! > > Consider, for example, the Cartesian product of the naturals with the > naturals, NxN = {(m,n} : m in N and n in N} with the well-order > (a,b) < c,d) > iff a < c or [ a=c and b < d ] > > Then for any n in N, the set of (a,b) for which a <= n is an initial > segment with no largest element, so there are as many such initial > segments without a largest as there are initial segments altogether. Look at what you quoted from me above. Where do I say anything about a last element? This is typical deliberate misinterpretation and a red herring. Thanks for the waste of time. Tony
From: David R Tribble on 7 Jun 2010 17:19 David R Tribble writes: >> Tony's ideas share a lot in common with those of >> Archimedes Plutonium (of which thousands of his posts >> can be found here in sci.math). I.e., the sequence of >> naturals (eventually) contains infinite naturals, infinite sets >> must contain at least one infinite member, all numbers >> must have digital representations, and a few other wacky >> ideas. > Jesse F. Hughes wrote: > What's so wacky about all numbers having digital representations? > Maybe you meant *finite* digital representations? AP does not accept anything as a "number" unless it has a digital representation, including (assuming that I am following his lines of reasoning, sometimes it's hard to tell what he's saying) numbers such as i (the imaginary unit). He also includes infinite integers having digital representations (alas, nothing consistent or coherent, of course). AP definitely has a hard time with numbers like w (omega) and Aleph_0, which he considers part of the "fake" standard math of the establishment. Throw in things like the quaternion i, j, and k, or primitive units for finite fields, as that's way off in the deep end for him. AP also has a hard time with irrationals, including his favorites e and pi, which are not "complete" numbers. Tony doesn't go that far, and to be fair to Tony, he knows a lot more actual mathematics than AP ever will. But but he does desire a digital representation for (real-based) numbers, including his "unit" infinities and infinitesimals. Tony's "T-riffics" are very similar to AP's "AP-reals". Both of them have a hard time providing a coherent, consistent, workable definition for them. (To other people's satisfaction anyway; they apparently don't see any problems on their side.) I said that both of their various ideas had a lot in common, but I didn't say they were all identical in every detail. -drt
From: MoeBlee on 7 Jun 2010 17:23 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: > > > > You cannot disagree with the fact that the rationals are a dense set > > > whereas the naturals are sparse, > > > WRONG, WRONG, WRONG. I've explained this to you several times, and you > > persist to ignore. > > > A set is dense with respect to some ORDERING. There IS a dense > > ordering of the set of natural numbers, and there is an ordering of > > the set of rationals that is NOT dense. > > > What you can say is that the set of natural numbers is not dense with > > regard to the usual ordering on the set of natural numbers and that > > the set of rational numbers is dense with regard to the usual ordering > > on the set of rational numbers. > > That's usually what I mean. I'll be more specific if not. 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. (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. 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. > > > 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? > > > 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. 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. MoeBlee
From: MoeBlee on 7 Jun 2010 17:27
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 |