Uncountability of the Rationals?
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From: Tony Orlow on 7 Jun 2010 20:54 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: > > > > > 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. Disingenuous. The "usual" ordering of a quantitative set is ... quantitative. That's what you mean by usual - ordered along "the line". > > (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. > 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. > > > > > 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. 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. > 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. > > MoeBlee- Hide quoted text - > > - Show quoted text - Peace, AppendixSpleen
From: Tony Orlow on 7 Jun 2010 21:01 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. With all the peace I can muster, TOny
From: Tony Orlow on 7 Jun 2010 21:02 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. Love, Tony
From: Tony Orlow on 7 Jun 2010 21:49 On Jun 7, 5:56 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 7, 7:59 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > On Jun 6, 5:31 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 5, 9:34 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > > On Jun 4, 3:16 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > As far as I know, neither Galileo nor Aristotle proposed what could > > > > fairly be called a set theory. > > > True, but they explored infinite sequences, Aristotle making the > > distinction between the two general types, and Galileo exploring > > bijection as a means of caomparison, and then conclusing that the > > sizes of infinite sets simply don't exist or cannot be calculated. > > (1) That doesn't address my point about their role in your analogy. I forgive you for missing the point. I love you anyway. > > (2) I'll take your word for it that that was Galileo's conclusion. I'm not an expert. I read that sort of recently. Feel free to correct me on that point. > > > the way, I only agree partly with Galileo's conclusion, but fully > > agree with the distinction between actual/uncountable and potential/ > > countable infinities. > > Whose distinction is that? It's not Aristotle's. As far as I know, > Aristotle's disctinction was between actual and potentential infinity, > as he endorsed the latter notion over the former. Yes, Aristotle doubted the existence of actual infinity but not the existence of the potentially infinite. To me, the actually infinite is distinct and extant. Things progress - no biggie. > > > > By the way, I should have mentioned that Orlow mentioned also Cantor > > > and Dedekind as counterparts of Jesus in Orlow's analogy. I don't see > > > how that would make sense in terms of his analogy, since Cantor and > > > Dedekind proposed uncountability, contrary to Orlow's own analogy. > > > If you really think that I don't believe in the distinction between > > the countable and uncountable then I don't know who you think you were > > talking to or what you think I said. > > You're reading into my remarks what is not in them. I didn't suggest > in the least that you decline a distinction between countable and > uncountable. Ummmm..... You say above, "Cantor and Dedekind proposed uncountability, contrary to Orlow's own analogy", which implies that I think that distinction between the two is part of the distraction, even though I've confirmed from my side of the aisle that it's a logically well founded distinction (as opposed to the well founding of a set), and not just simply logically consistent. > > > Additionally, it is not ZFC that > > I object to, but the extension of the theory with the model or > > cardinality as "set size" for the infinite case. Simple bijection does > > not equate to "equinumerosity" in my book. > > This is where AGAIN you just skip recognizing my arguments and > explanations (SEVERAL of them over years) about this. It seems you > don't even know what I'm talking about, no matter how carefully, > simply, or, in contrast, fully explain. I surmise you just skim right > past. You aren't telling me anything new. I've heard it. If you want a response, then say something which addresses the problem which I haven't heard before. I isn't dumbses. Tweren't mees. > > > That's "equicardinality" > > for the mathematically rigorous, no? > > You're welcome to make up the word. In the meantime, 'equinumerous', > 'equipotent', 'equipollent' are commone enough. Then, with cardinals > incorporated, we can also say 'having equal cardinality', etc. Equispluperous, or whatever. That's yer rigger fer ya. Yep, Maffomaticol rigger. I readsed dat on de insternet. Farscinating! "Equicardinal" is sufficient. > > > The analogy has to do with what Hilbert et al did with ZFC, and what > > Constantine did with Christianity. > > I got that more or less. But the Jesus mention in the analogy was then > left dangling. Not a big deal; I just wanted to see where you'd land > with it. I never pretended that Jesus was the Son of God. I don't even usually spell god that way, with a CAP. He was treasured son of the many, learned and taught and was Rabbi. You take your gems from the earth and feel lucky for them. I land on my feet, or the tiger links away, to lick his wounds. The cliffs are predictable this season. There is something in the air. (I'm a little buzzed. I have that right occasionally) :) > > > > > > Anyway, so everything Galileo and/or Aristotle said was true and wise? > > > > > So _nothing_ Galileo and/or Aristotle said was true > > > > and wise? > > > > Of course Galileo and Aristotle were wise in certain ways. So what? > > > So were Abraham and Buddha, who each contributed to the wisdom of > > Jesus. That's what. > > That's a fine non sequitur. No, it is a tadbit of information which probably isn't in your repertoire. Of the 18 years missing from the western account, most were spent with hindus and buddhists, leanring how to play dead, having traveled on one of uncle's three boats to the East and back. But, anyway.... > > > > > > > ToeKnee > > > > > BoerEeng. > > > > > "Boring." This, of course, represents the reaction > > > > that MoeBlee would have if TO really were to post a > > > > rigorous theory applicable to sciences based on his > > > > own non-ZFC notions. MoeBlee would be "bored." > > > > No it doesn't. STOP PUTTING WORDS IN MY MOUTH. > > > > If Orlow ever came up with a rigorous theory I'd be amazed and > > > fascinated. STOP PUTTING WORDS IN MY MOUTH. > > > Don't misrepresent my position and it won't come back to bite you. > > What are you talking about? I've not intentionally misrepresented you, > and when I have inadvertently, I've been diligent to correct myself. > That is in stark contrast with Transfer Principle's posting about me. Well, I don't know too much about your relation with Transfer, but mine is going relatively well. Anyway, now that I tok a break, I remember that your comment here was originally about My "ToeKnee" thang, and not aboput anything subtantial, thus the "FootLeg" signature. <3 > > > > The point I made by saying "BoerEeng" is only that Orlow has worn out > > > whatever amusement there ever was by signing himself with the > > > affectation "ToeKnee" in response to my posts signed "MoeBlee". > > > My ApoloGees. hyuck hyuck. The funny thing is that it bothers you at > > all. > > Not as funny as that you're so pleased with your repetitiveness. What'd you say, agin'? > > > FootLeg > > 'FootMouth' would suit you better. > > MoeBlee Y-not evoL
From: Tony Orlow on 7 Jun 2010 21:54
On Jun 7, 7:54 pm, 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. > > 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 embue it with an > ordering. > > -drt Why are all of these infinite sets you cite some kind of sequence? Me no stand under it. Only have coco nut, and this rock. Also I find stick. Sorry for the sarcasm... Formulae ams there for they reasons. Algebra is a good gift. Tony |