An uncountable countable set
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From: Randy Poe on 27 Sep 2006 10:37 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Virgil wrote: > >>>>> In article <45189d2a(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>> Virgil wrote: > >>>>>>> In article <45187409(a)news2.lightlink.com>, > >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> If so, then why do you say "There is no > >>>>>> infinite case"? > >>>>> Because there isn't any. > >>>> There is no noon in the Zeno machine? > >>> There's no ball put in at noon in the Zeno machine. > >>> > >>> All cases, all Zeno balls, are put in before noon. > >>> > >>> - Randy > >>> > >> At any time finitely before noon, only a finite number of balls have > >> been processed. Do you disagree? > > > > I agree. > > > >> Are you suggesting that the vase fills > >> at some *infinitesimal* amount of time before noon? > > > > No. There is no time before noon when the vase is full. > > Let t be any time before noon. Infinitely many balls > > will be inserted into the vase after t, but before noon. > > > >> How does that differ, in your view, from being *at* noon? > > > > Well since I also don't say "it fills at an infinitesimal > > time before noon", there is no need to distinguish > > one nonsensical statement from the other. > > > > But, you do say it fills, Probably not. I wouldn't know how to describe "full" for an infinite vase. > or empties, right? It definitely empties, since every ball you put in is later taken out. > And, at the same time you > say it does not do so at noon, nor does it do so before noon. When does > this occur? It doesn't. There is no last ball, so I wouldn't say there's a point where the nonexistent last ball goes in, nor would I say there's a full state corresponding to the nonexistent last ball. So the reason I don't say it's full "an infinitesimal time before noon" or "some other time before noon" is that I don't say it's full. - Randy
From: Tony Orlow on 27 Sep 2006 10:41 Virgil wrote: > In article <45193afa(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <45189e62(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <45187864(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Han de Bruijn wrote: >>>>>>> Tony Orlow wrote: >>>>>>> >>>>>>>> Mike Kelly wrote: >>>>>>>> >>>>>>>>> Han de Bruijn wrote: >>>>>>>>> >>>>>>>>>> Mike Kelly wrote: >>>>>>>>>> >>>>>>>>>>> Han de Bruijn wrote: >>>>>>>>>>> >>>>>>>>>>>> Mike Kelly wrote: >>>>>>>>>>>> >>>>>>>>>>>>> What the hell are you talking about? Arguing with someone who >>>>>>>>>>>>> can't >>>>>>>>>>>>> speak English is getting aggravating. >>>>>>>>>>>> My English is much better than your Dutch. >>>>>>>>>>> So what? Your English is still too poor for this discussion to be >>>>>>>>>>> fruitful. >>>>>>>>>> Still don't get the point, huh? >>>>>>>>>> >>>>>>>>>> You are lacking even the most elementary form of politeness. It's >>>>>>>>>> very >>>>>>>>>> impolite to cut of a discussion with somebody from a foureign >>>>>>>>>> country - >>>>>>>>>> somebody who is doing his best to communicate with you - only >>>>>>>>>> because >>>>>>>>>> you are obviously superior in expressing your thoughts within your >>>>>>>>>> own >>>>>>>>>> mother's tongue. >>>>>>>>> You're a very rude person yourself, Han. I generally don't feel the >>>>>>>>> need to be civil to those who won't reciprocate. >>>>>>>> I don't think I have ever found Han to be rude, except when he >>>>>>>> referred to my "babbling" recently. Ahem. But anyway, while we >>>>>>>> disagree on the actuality of any infinity, we have the open mind of >>>>>>>> spirited debate, and feel no need to get nasty. >>>>>>> I may be rude sometimes, but I never get _personal_ by calling somebody >>>>>>> an "idiot" or a "crank". Tony's "babbling" translates with Euroglot as >>>>>>> "babbelen" in Dutch, which is a word I can use here in the conversation >>>>>>> with my collegues without making them very angry (if I say "volgens mij >>>>>>> babbel je maar wat"). But, of course, I cannot judge the precise impact >>>>>>> of the word in English. Apologies if it is heavier than I thought. >>>>>> Do you have the saying, "Shallow brooks babble, and still waters run >>>>>> deep"? I figured you picked up the usage from this forum, actually. It's >>>>>> meant, in English, to mean you aren't making any sense. :) >>>>>> >>>>>>>> Furthermore, I have never had any trouble understanding what Han is >>>>>>>> saying, except where he is using some mathematical construct with >>>>>>>> which I am not familiar. His English is not bad, and blaming your >>>>>>>> disagreement on his inability to communicate is kind of low. >>>>>>> Thank you very much, Tony, for this sort of defense. >>>>>> My pleasure. It seemed like a vacuous excuse. I get pretty sick of those >>>>>> diversionary tactics. >>>>>> >>>>>>>> So, let's engage in lively debate, and maintain our civility, while >>>>>>>> chopping each other's arguments to pieces. Of course, this can only >>>>>>>> happen if we don't consider our arguments to be part of our anatomy. >>>>>>>> Otherwise, it gets personal. >>>>>>>>>>> You are misinterpreting virtually all my posts. You claim that >>>>>>>>>>> you're >>>>>>>>>>> not dishonest so I have to conclude you're simply incapable of >>>>>>>>>>> comprehending written English. This makes this whole subthread >>>>>>>>>>> pointless. >>>>>>>>>> I have only this kind of trouble with _you_ and nobody else on the >>>>>>>>>> web. >>>>>>>>> Really? You've never had anybody else other than me complain that you >>>>>>>>> misinterpret their posts? I suppose I must have hallucinated dozens >>>>>>>>> of >>>>>>>>> posts I've seen of just that, then. >>>>>>>>> >>>>>>>>> You've never had anyone other than me struggling to understand what >>>>>>>>> the >>>>>>>>> devil you mean by your broken English? I must have hallucinated, for >>>>>>>>> example, "A little physics would be no idleness in mathematics", then >>>>>>>>> :)? >>>>>>>> Well, that's a difficult type of quote. Han - I wouldn't mind working >>>>>>>> on exactly how you want to say that in English, if you like. :) >>>>>>> Uhm, since litteraly everybody is complaining ... Let it be an >>>>>>> encrypted >>>>>>> message then :-) >>>>>> Well, it seems to me that perhaps you're saying something like, "Those >>>>>> with their heads in the abstract should keep their feet in the >>>>>> concrete", though that sounds a little funny. >>>>>> >>>>>> Math=Science? >>>>> Scientists, particularly those in the sciences most dependent on >>>>> mathematics, tend to think that all mathematics is, or should be, a >>>>> subservient to their particular fragment of science. >>>>> >>>>> Mathematicians know better. >>>> Define "better". Those that work in various areas of science share a >>>> notion which defines science. Theories which have no means of >>>> verification are not science, but philosophy. In mathematics, >>>> verification really consists of corroboration by other means, agreement >>>> between different approaches. In science, where you find a contradiction >>>> with your theory, it needs revision. So, the scientific approach to >>>> mathematics requires some criterion for universal consistency, as >>>> measured by the predictions of the various theories that comprise it. >>>> Where two theories collide, one or both is in error. I think that's >>>> better. >>> TO mistakes misapplication of mathematics, which is an error by >>> scientists, as an error of the mathematics. >>> >> That's not what I said, and you know it. > > When an application of mathematics to physics, or some other science, > does not predict what actually is observed in that science, why does TO > insist the problem is inherent in the mathematics alone and not in the > improper application of mathematics to science? It's not a matter of what's observed in physical science. I am saying that the general method of science can be applied to mathematics, where verification of a theory is through agreement with other theories as far as conclusions are concerned. Where two theories collide, and both seem sound, there is usually a generalization which includes both as parts, right? Like Euclidean geometry is now a category
From: Tony Orlow on 27 Sep 2006 10:48 Virgil wrote: > In article <45193e6f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Han de Bruijn wrote: >>> Tony Orlow wrote: >>> >>>> Virgil wrote: >>>>> Mathematicians know better. >>>> Define "better". Those that work in various areas of science share a >>>> notion which defines science. Theories which have no means of >>>> verification are not science, but philosophy. In mathematics, >>>> verification really consists of corroboration by other means, >>>> agreement between different approaches. In science, where you find a >>>> contradiction with your theory, it needs revision. So, the scientific >>>> approach to mathematics requires some criterion for universal >>>> consistency, as measured by the predictions of the various theories >>>> that comprise it. Where two theories collide, one or both is in error. >>>> I think that's better. >>> Precisely ! In mathematics, there are contradictory approaches, such as >>> constructivism (Brouwer) against axiomatism (Hilbert). Its practicioners >>> are asked to be "nice" to each other and to "reconciliate" the different >>> points of view, which turns out to be a hopeless task. Such a situation >>> would be unthinkable if mathematics aimed to be a science. >>> >>> Han de Bruijn >>> >> Well, Han, I'm not sure I agree with the statement that reconciliation >> is hopeless. Is it hopeless to reconcile the wave nature of elementary >> entities with their particle nature? > > It is close to hopeless to expect those who reject the law of the > excluded middle (constructionists) and those who insist on it > (formalists) to agree. > If neither can appreciate the other's point, perhaps. Some christians get along quite well with some muslims. The question boils down to whether 0^0 is 1. If so, constructivists are up a creek without a paddle. If it can be considered to be 0, then they have a point. Can it? > > >> There is confusion about my "definition" of infinitesimals, because I >> can see the validity both in nilpotent infinitesimals and in those that >> are further infinitely divisible. > > Until TO can come up with an axiom system which simultaneously allows > his infinitesimals to be both nilpotent and not, he is in trouble. > For purposes of measure on the finite scale, infinitesimals can be considered nilpotent. That's all. Do you disagree? > >> Constructivism and Axiomatism are two sides of a coin. They can be >> reconciled in larger framework, I think. > > Better men that TO will ever be have tried. I am sure there are better men trying right now. Are you one of them? :)
From: Tony Orlow on 27 Sep 2006 10:50 MoeBlee wrote: > Tony Orlow wrote: >> Constructivism and Axiomatism are two sides of a coin. They can be >> reconciled in larger framework, I think. > > I don't know what your definition of 'axiomatism' is, but there are > axiomatic systems for constructive mathematics. > > MoeBlee > I dunno. I was responding to Han's comment. I think he means constructive concepts vs. axiomatic declarations. ToeKnee
From: Tony Orlow on 27 Sep 2006 10:51
David R Tribble wrote: > Tony Orlow wrote: >> For the sake of this argument, we can talk about infinite reals, of >> which infinite whole numbers are a subset. > > What are these "infinite reals" and "infinite whole numbers" that you > speak of so much? > > If you've got a set containing the finite naturals and the "infinite > naturals", how do you define it? N is the set containing 0 and all > of its successors, so what is your set? > The very same, with no restriction of finiteness. Any T-riffic number has successor. :) Tony |