An uncountable countable set
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From: Lester Zick on 24 Oct 2006 14:02 On 23 Oct 2006 11:34:32 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> You and he just have different perspectives on the problem and nothing >> in what you have to say has any relevance to what Tony believes any >> more than what Tony believes has any relevance to what you believe. > >Differing perspectives are welcome. But that is different from simply >saying incorrect things about the technical points and also from giving >the kind of woozy arguments he gives for his non-axiomatic mathematics. I don't necessarily agree with Tony, Moe. In fact to be blunt about it I disagree with most of what he is suggesting just as he disagrees with me. The difference is that I find his search for mathematical meaning considerably more flexible and tolerant of disagreements with establishment views of set analysis and techniques. The point is that if orthoxy endorses contradictory notions (and here I'm primarily thinking of violation of containment, what are generally called counter intuitive results, but I'm confident there are others as well) it can scarcely complain when others embrace comparably contradictory notions to do whatever they see fit. So your arguments for establishment views of set analysis just don't carry any weight with me when you can't prove that they're true to the exclusion of other set analytical perspectives. Nor do I much care what kind of buzzword rationalizations and definitional techniques standard set analysis uses to justify its own perspectives. It's all just a smoke screen as far as I'm concerned, so much jargon and verbiage used to simulate a sophisticated technical mathematical edifice where there is none. ~v~~
From: Lester Zick on 24 Oct 2006 14:40 On 23 Oct 2006 11:00:30 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> On 19 Oct 2006 14:35:08 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >> >Lester Zick wrote: >> >> And if set theorists ever get around to formulating a mathematical >> >> theory do let us know. >> > >> >There are a whole bunch of recursively axiomatized mathematical >> >theories that are set theories. Z set theory (and its variants) has >> >been in good stead as an axiomatic theory since Skolem mentioned how to >> >handle the previously too vague notion of a definite property. >> >> I have no idea what you think mathematics is, Moe, > >I can offer you many texbooks, book, articles, and journals that are >just the beginning of the mathematics I would like to study. Meanwhile, >I asked you for just a single reference to what you study. You replied >by saying there is none (or words to that effect). I gave you the only reference I had to my own "Epistemology 201: The Science of Science" which so far as I know is the only demonstrable framework for the exhaustion of truth ever achieved. I don't know what else to say except that mathematics and science generally represent the search for and canonization of truth in demonstrably, exhaustively reduced terms and that apart from that we might just as well all pack it in and practice variants of speculative pythagorean metaphysics. >> but when you use >> phrases like "recursively axiomatized" they sound more like slogans >> than mathematics. > >'recursively axiomatized' is a precisely, rigorously defined predicate >in mathematics. I can't help that it sounds to YOU like a slogan, since >I can't help that you've never studied the subject. Why should one study an edifice of assumptions recursively axiom atomized (sic) or not when what matters is whether the axiomatic assumptions are demonstrably true and exhaustive and not whether they're rationalized six ways to Sunday. What I see in modern math is a wholesale substitution of terminological regression for mathematical explanation. That's what I call jargon. Every time someone reaches for an explanation they just come up with another word to describe the same thing in a somewhat more obscure way. Then they pretend that without that particular obscurity we could never employ very easily defined concepts like cardinality, sets, arithmetic and so on. >> Others that come to mind are "robust" and "well >> ordered". > >I didn't use the word 'robust'. But 'well ordered' is a precisely, >rigorously defined mathematical term. No I understand. I used "robust". But I've seen every bit as many references to "robust" theories as to "well ordered" sets. At least to me the "well" in "well ordered" is redundant. If a set is ordered it is "well ordered" and if not it's not ordered. Of course as you note I haven't studied the arcana of conceptual techniques in standard set analysis so I may be mistaken but that's how it appears. >> I'm confindent sets and set methodologies have numerous >> uses. I just can't tell what they might be from anything you've had to >> say. All I hear are things like the "axiom of regularity" the "axiom >> of infinity" and the "axiom of choice" which sound more like buzzwords >> used purely to justify things no one can demonstrate true or false. > >They are formal axioms - precise formulas. They're definitely not just >buzzwords, except as used by cranks. But the point is that they're all assumptions. Otherwise they wouldn't be axioms and we could easily study and classify all these assumptions in exhaustive terms instead of ambiguous nomenclature. The formality and precision of definition are irrelevant when we have no way to say whether they're true in mechanically exhaustive terms (especially considering the atrocious manner of accepted mathematical definition). Dressing up mathematical concepts like a dogs dinner with obscure terminology doesn't make them true and exhaustive and that's primarily what math and science are or should be after. ~v~~
From: Virgil on 24 Oct 2006 16:41 In article <453e3ed0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > So, David, you think the fact that balls leave the vase only by being > removed one at a time, and the fact that at all times before noon there > are balls in the vase, and the fact that at noon there are no balls in > the vase, is consistent with the fact that no balls are removed at noon? Quite so! > How can you not see the logical inconsistency of an infinitude of balls > disappearing, not just in a moment, but at no possible moment? The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? > Are you > so steeped in set theory that you cannot see that an unending sequence > of +10-1 amounts to an unending series of +9's which diverges? What is > illogical about that? The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? > > In your set-theoretic interpretation of the experiment there is a > problem which makes your conclusion incompatible with conclusions drawn > from infinite series, and other basic logical approaches. It is not that > I don't understand how your logic works. It's that I see clearly that it > doesn't, and I'm trying to precisely pin down exactly where the error > is. The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? > It's not an easy task, since this transfinite theory is rather well > crafted and tweaked over the years. However, there are clear reasons, > once the matter is fully investigated, why the logic fails. The > conclusion produces clear contradictions in terms of a time of emptying > and the requirement at some point of a negative number of balls in the > vase in order for it to empty at all, and it all derives from using the > Zeno schedule to complete a sequence which has no end, hiding this fact > in a time singularity at t=0. The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? > > Very basic logic would hold that, if the vase is not empty at any time t > such that -1<=t<0, and the vase is empty at t=0, then balls were removed > at t=0, since that's the only way the vase can become empty. However, > t=0 corresponds, according to the stated schedule, to infinite index n > in the sequence, and an infinite label on a ball, which is not allowed, > as per the experiment. Therefore, no ball can be removed at t=0, and the > vase cannot become empty at that point, or at any point before. > > I asked you when you thought the vase became empty. You avoided the > question, saying it was interesting, and then going on with your same > tired formulation of the problem, as if I haven't followed the logic and > pointed out the flaw in the approach. > > So, answer the question. When does this miracle of emptiness occur? The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer?
From: Virgil on 24 Oct 2006 16:41 In article <453e3f1a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > cbrown(a)cbrownsystems.com wrote: > >> stephen(a)nomail.com wrote: > >>> With the added surreal twist that the limit of the number > >>> of unnumbered balls in the vase as we approach noon is 0, > >>> but the number of unnumbered balls in the vase at noon is > >>> infinite. :) > >> I think his response, when I pointed this out to him, was either "Oh, > >> shut up!" or "Whatever." > > > > That is consistent with my suggestion that Tony is reasoning by > > imagining a vase filling up. If you visualize the vase filling up in > > your mind, you don't see the unnumbered balls in the picture. > > > > If you have an infinite ocean wit 10 liter/sec flowing in, and 1 > liter/sec flowing out (and no evaporation), will it ever empty? No. Same > difference. The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer?
From: Virgil on 24 Oct 2006 16:43
In article <453e3fe7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <453d52d1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > For it to "become" anything implies a gradual and continuous change > > spread over some time interval of positive length, which is not the case > > here. > > Not if removals occur infinitely quickly. It can happen in a moment. > But, there has to be at least one moment involved in the event, or it > didn't happen. That moment can't be before noon. It can't be noon. Is it > after noon, and then time-travels back to happen after all the moments > before noon, but just in time to beat noon? You have pretzel time, Virgil. > > > > > The vase is non-empty at every time before noon and empty at noon in > > much the same discontinuous way that Sign(x) is non-zero at ever real x > > except x = 0. > > Sign(x)? Whatever. y=9x diverges as x->oo. Sorry. The only relevant question is "According to the rules set up in the problem, is each of the balls inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? |