Uncountability of the Rationals?
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From: Virgil on 8 Jun 2010 15:43 In article <62bb8123-11c6-458d-9d10-09764deb196a(a)x21g2000yqa.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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 S1 and S2 may be regarded of as sequences (provided the relative position of each entry is taken as argument for the function providing the value in that position). But even in that limited sense, neither S3 nor S4 is a sequence.
From: Virgil on 8 Jun 2010 15:49 In article <877f1271-61d5-4c42-9590-555117e640e4(a)y4g2000yqy.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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? The only difference I can see is that "a countably infinite complete list of digital numbers" is sufficiently ambiguous as to be meaningless where "snarflek" is merely undefined. > > > > > > 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. Neither so vacuous nor so spewagelike as most of TO's presentations.
From: Virgil on 8 Jun 2010 15:55 In article <9cebfe8f-89f8-428c-8df7-f961eb665080(a)i28g2000yqa.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 8, 10:11�am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 7, 11:38�pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > 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? > > > > WHAT exact ordering on sets in GENERAL do you mean '<' to stand for? > > > > MoeBlee > > I was referring to sets of real (or hyperreal) quantities, > specifically. Changing the subject doesn't prove anything. Then stop doing it! And if you really were referring only to such sets, you should have said so instead of speaking as if all sets were allowed. TO again being sloppy, and then getting pissed when called on it.
From: Transfer Principle on 8 Jun 2010 18:03 On Jun 7, 2:23 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 5, 12:44 am, Tony Orlow <t...(a)lightlink.com> wrote: > > 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. Here's what I believe about TO's Bigulosity. MoeBlee refers to extrapolating from the finite to the infinite when considering set size. Here are two equally intuitive notions that describe set size for finite sets: 1. Sets in bijection with each other have the same size. 2. The whole is strictly greater than the part. (paraphrased from Euclid) The problem is that for (Dedekind-) infinite sets, there is no example of set size that preserves both of these notions, since by definition there exists a bijection between a D-infinite set and one of its proper subsets. So the best we can do is choose one of these two equally good notions to preserve. Standard Cantorian cardinality rejects the second in favor of the first. But I see no reason that we can't reject the first in favor of the second -- and this would give us TO's Bigulosity. I'd love to believe that there can be a completely rigorous theory applicable to the sciences in which the first notion is rejected for the second. I see no reason for the lack of symmetry in that there's a sensible set size that maintains the first property but none that maintains the second property. We know all about standard cardinality. I want to know how to develop a set size that preserves the second property of finite set size.
From: MoeBlee on 8 Jun 2010 18:41
On Jun 8, 5:03 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > 1. Sets in bijection with each other have the same size. > 2. The whole is strictly greater than the part. > (paraphrased from Euclid) Yes, this is the ages old dilemma. Galileo's paradox, so to speak. We all know about it. > So the best we can do is choose one of these two equally > good notions to preserve. Standard Cantorian cardinality > rejects the second in favor of the first. But I see no > reason that we can't reject the first in favor of the > second -- and this would give us TO's Bigulosity. Fine. Except "bigulosity" awaits a coherent definition. Among the irritations provided by Orlow is that he acts AS IF he's given adequate definition and treatment of all his ersatz terminology while the rest of us are just a bunch of dopes, dupes, or knaves who can't or won't recognize the wonderfulisity of his "bigulosity". MoeBlee |