Uncountability of the Rationals?
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From: MoeBlee on 8 Jun 2010 09:36 On Jun 7, 8:01 pm, 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. > I don't even know the basics? That's what I said. You 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. You mention this as a demonstration that you know the basics? The continuum hypothesis is independent of the axioms. So? There is NO consistent theory adequate for "a certain amount of" arithmetic that is without independent sentences. ZFC proves there is a well ordering of the reals. Also we have a result (I"m stating roughly here) that there is no explicit formulation of a well ordering of the reals. Admittedly that seems to be an odd state of affairs and I wouldn't begrudge someone for being dissatisfied with it. However, it is not in itself a contradiction. > Remember my big challenge? Your "big challenge". Sounds like a bad action movie or something. > What > is the width of a countably infinite complete list of digital numbers? What "list" are you referring to? There IS NO countable list of all the denumerable sequences on an alphabet with cardinality greater than 1. And "width" in this context requires a DEFINITION from you. > 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. Please stop wasting my time with your babble. The above is almost as bad as Finlayson. MoeBlee
From: MoeBlee on 8 Jun 2010 10:08 On Jun 7, 8:49 pm, Tony Orlow <t...(a)lightlink.com> wrote: > 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. I got your point the first time. You just missed mine again. I allow that you're a human being with rights 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. No, as I said, I'm not informed on that particular matter. > > > 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. Yes, as I just said, Aristotle endorsed potential infinity as opposed to actual infinity. Countable and uncountable did not enter into his reckoning, as far as I know. > 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. Okay, fair enough. I'll retract the point on uncountability itself. But the problem still remains that if you endorse Cantor, then it really makes no sense to object to ZFC (or to what you call "extension" of it to include a notion of 'size' or whatever) since ZFC and the set theoretic notion of size are SUBSUMED by Cantor. > > > 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. You've never responded substantively in the first place, when I presented the matter anew. Now it's like this: Tony's mommy says, "Tony, you broke the coffee table in the living room. Please replace it." Then no answer from Tony even after his mommy has been mentioning the matter over and over. Finally, Tony does answer: "I've already heard you about this. If you want me to respond, then please tell me something new." Great logic, Tony! > > > 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! Please stop babbling. > > > > > "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. Fine, then your remark "Don't misrepresent my position and it won't come back to bite you" was mere more gratuitous blabber. MoeBlee
From: MoeBlee on 8 Jun 2010 10:11 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
From: MoeBlee on 8 Jun 2010 10:17 On Jun 8, 7:45 am, Tony Orlow <t...(a)lightlink.com> wrote: > 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. "Greater quantity" defined for SETS IN GENERAL HOW? What specific ordering on SETS IN GENERAL do mean? (And I don't even ask you to state it explicity, but even just to prove that there exists some unique specific ordering that we can name in order to define 'the usual ordering' even if not describe.) Please stop here. Address this. Put up or shut the Foeck Oep please. MoeBlee
From: MoeBlee on 8 Jun 2010 10:19
On Jun 8, 7:52 am, Tony Orlow <t...(a)lightlink.com> wrote: > Sorry, Tim, but I'm unlikely to respond to any more of your comments, > since they are simply vacuous spewage. A veritable BP oil rig of spewage calling the other poster black! MoeBlee |