An uncountable countable set
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From: Tony Orlow on 3 Oct 2006 03:33 Virgil wrote: > In article <4521fcd5(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <1159798407.217254.275710(a)m7g2000cwm.googlegroups.com>, >>> mueckenh(a)rz.fh-augsburg.de wrote: >>> >>>> Tony Orlow schrieb: >>>> >>>> >>>>>> This is an extremely good example that shows that set theory is at >>>>>> least for physics and, more generally, for any science, completely >>>>>> meaningless. Because the numbers on the balls cannot play any role >>>>>> except for set-theory-believers. >>>>> Yes. I was flabbergasted by this example of "logic". The amazing thing >>>>> is, set theory is supposed to apply where we know nothing except for the >>>>> membership status of each element in a set, and yet, here is applied >>>>> this property of labels that set theorists claim is crucial to answering >>>>> the question. Set theory in the finite sense is a fine thing, but when >>>>> it comes to the infinite case, set theorists don't even know anymore >>>>> what they're TRYING to do. >>>> One should think that set theory, if useful at all, should be capable >>>> of treating problems like this. But here we see it fail with >>>> gracefulness and mastery. >>>> >>>> Set theorists always see only the one ball escaping the vase but not >>>> the 9 remaining there. So they can accept that there are as many >>>> natural numbers as rational numbers. I cannot understand how this >>>> theory could invade mathematics and how I could believe it over many >>>> years without a shade of doubt. >>>> >>>> Regards, WM >>> If we alter the problem to start with all the balls in the vase and >>> remove them according to the original schedule then every ball spends at >>> least as much time in the vase as before, but not everyone will see that >>> the vase is empty at noon. >>> >>> Those who argue that having the balls spend less time in the vase leaves >>> more of them in the vase as noon, have some explaining to do. >> No, if you start with all the balls in the vase, it clearly becomes >> empty, but when you are adding more balls per iteration than you are >> removing, the eventual emptying of the vase is impossible. > > Thus TO claims that by putting balls into the vase earlier, but removing > them as before, one will find fewer of them in the vase at noon. > > I do not see the logic of that argument. Because, in your version of a mind, you have decoupled the addition and subtraction of balls, so that they no longer form the same sequence of events as originally stated.
From: Virgil on 3 Oct 2006 03:36 In article <4522085d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > Virgil wrote: > >>> Except for the first 10 balls, each insertion follow a removal and with > >>> no exceptions each removal follows an insertion. > > > > Tony Orlow wrote: > >>> Which is why you have to have -9 balls at some point, so you can add 10, > >>> remove 1, and have an empty vase. > > > > David R Tribble wrote: > >>> "At some point". Is that at the last moment before noon, when the > >>> last 10 balls are added to the vase? > >>> > > > > Tony Orlow wrote: > >> Yes, at the end of the previous iteration. If the vase is to become > >> empty, it must be according to the rules of the gedanken. > > > > The rules don't mention a last moment. > > > The conclusion you come to is that the vase empties. As balls are > removed one at a time, that implies there is a last ball removed, does > it not? No. Not if infinitely many can be removed in finite time. > > > > The rules state that for each point in time, at 1/2^n seconds prior to > > noon, 10 balls are added to the vase, and then the ball that was > > previously inserted earlier than all the others is removed. (I.e., at > > time 1/2^n, balls 10n+1 thru 10n+10 are added, and ball n is > > removed.) > > > > Therefore the rules stipulate that balls are added and removed > > at each moment 1/2^n sec before noon, for each n = 1,2,3,... . > > > > By assuming that there is some last moment when the vase is emptied, > > or when the last ball is added, or whatever, you are assuming that > > there is a largest n. That's your assumption, because it's not > > mentioned in the rules. But it's an unwarranted assumption, by the > > simple fact that there is no largest n (as you yourself have proclaimed > > many times). > > > > [Personally, I think the problem is a bit simpler if only two balls > > are added and one removed at each step. But, whatever.] > > > > Sure, but that doesn't fix the problem with adding more than you're > subtracting and coming up with an empty till. if TO were right, then TO should be able to resolve the problem by showing which balls remain at noon.
From: Virgil on 3 Oct 2006 03:42 In article <4522094d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > In article <1159649021.675137.307470(a)k70g2000cwa.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > > > So, again, no definition. Where did I not speak the truth? > > > > > > > > > > Here: "...because he never answered to questions about it". > > > > > > > > You gave on Usenet a definition of natural number in answer to my > > > > question > > > > for it. I posted questions about that definition, and you never > > > > answered > > > > them. So in what way is it a lie when I state that you never answered > > > > them? > > > > > > I did not see an unanswered question. > > > > Apparently you did not see the questions at all. You stated as your > > definition > > something like: > > "numbers are in trichotomy with each other" > > and I answered something like "so omega is a number". But I never did see > > an > > answer. > > Is omega greater than any finite number? Unless it is also a number, it cannot be compared with numbers, finite or otherwise, so that TO is implying it is a number by asking that question. > > > > > I asked for a mathematical definition of number. You never gave one (and > > also your paper does not give any mathematical definition), your only > > mathematical definition was about the trichotomy. > > That's an important question. Can you answer it? When TO has answered what he has been asked, he will be in a better position to ask questions. > > Variables do exist, but are not numbers. > > They represent numbers, and what are numbers, but representations of > quantity? Every number is a set in ZF or in NBG. If TO wants them to be anything else, he needs to give us his axiom system first.
From: Han de Bruijn on 3 Oct 2006 03:42 Dik T. Winter wrote: > In article <1159726829.184763.303470(a)i3g2000cwc.googlegroups.com> > Han.deBruijn(a)DTO.TUDelft.NL writes: > > stephen(a)nomail.com schreef: > > > > > I suppose I should clarify this. You can approach the infinite > > > using the the limit concept, but you always have to be careful > > > when using limits, and you have to be precise about what you > > > mean by the limit. > > > > Okay. But the point is whether there exist infinities that can _not_ > > be approached using the limit concept. > > Yes. In ordinals they are called the "limit ordinals". The name is not > very appropriate, I think, because they are the ordinals that defy the > limit concept, but that is just naming. > > > Obviously they exist, because > > how can we approach e.g. the Continuum Hypothesis by employing limts? > > I have no idea about the meaning of this statement, but off-hand I ould > say that there is no way. Now we are getting somewhere. In applicable mathematics, approaching the infinite via the limit concept is the _only_ viable way. All the rest is nonsense (= has no "sense": can never be measured with a sensor). Han de Bruijn
From: Virgil on 3 Oct 2006 03:45
In article <45220ad7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > But that is not their "natural" order, and TO elsewhere insists that we > > follow natural orderings. > > From Dik Winter, elsewhere in this thread: > > "But that has ordinal k + omega, not -k + omega. A set of k negative > numbers has ordinal k; not -k." But To insists that the reals , and thus all subsets, take the order of the complete ordered field of reals. |