Cantor Confusion
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From: David R Tribble on 9 Oct 2006 22:46 William Hughes schrieb: >> "For any natural N, the ball numbered N will be removed from >> the vase before noon" > MueckenH wrote: > There is not "any natural" but only those which we can define. There is > a largest natural which ever will be defined. Hence mathematics in the > universe and in eternity has to do with only a very small sequence of > naturals. Let's give that largest natural a name: M. So the largest natural which will ever be defined (presumably by the number elves that define all the numbers that we humans use) is M. Then, by your definition, M+1 cannot be a number. We can't even talk about it or hypothesize anything about it, because it's just too large to be a number, and the universe won't exist long enough for anyone to define it (even though you said that the numbers exist in eternity - I guess that's a short eternity, not the long kind?). I wonder, though, is M odd or even? Of course, I can't speculate if M+1 is odd or even, though, because it's been defined that I can't. > Writing 1,2,3,... is but cheating Which of your rules does it break?
From: imaginatorium on 10 Oct 2006 02:31 Dik T. Winter wrote: > In article <1160377400.288823.275240(a)c28g2000cwb.googlegroups.com> cbrown(a)cbrownsystems.com writes: > > Dik T. Winter wrote: > ... > > > The balls in vase problem suffers because the problem is not well-defined. > > > Most people in the discussion assume some implicit definitions, well that > > > does not work as other people assume other definitions. How do you > > > *define* the number of balls at noon? > > > > I disagree. > > Do you? > > > This is no more difficult than asking "how do you *define* the number > > of balls at pi/10 seconds before midnight?" > > And you start giving a definition. I do not say that it is *difficult* to > define. But in my opinion more than one definition is possible. I agree with Chas - this simply isn't anything to be _defined_. Of course the problem as stated is in real-worldy-looking terms, as though this were plausibly an experiment one could do (can you say "intuition pump"), and it is necessary to _define_ how the statement is to be interpreted mathematically. This seems to be fairly simple (forgive me if I use slightly the 'wrong' formula, corresponding to my "sliver" example, but it makes no real difference): For every natural number n there is a ball, and this ball has two states along a timeline (value represented by t): When -2/n <= t <= -1/n state is IN When t < -2/n or t > -1/n state is OUT If the question is: For how many of these balls is the status IN at t=0, the answer is completely clear: none at all. No further "definition" is needed, or even possible. The cranks universally proceed from some intuitions about the real world that - essentially - there are no discontinuous functions in physics. If there are no discontinuous functions, then it follows that you can "swap limits" - and in particular that if balls(t) is a function representing the number of balls in state IN at time t, then lim t->0 (balls(t)) = balls(0). If there _are_ discontinous functions, this clearly does not follow. I do find it hilarious that one of the cranks proudly displays a graph of y = 1/x on his website, and seems to believe this tells us "the value of 1/0". Well, a little education would be no idleness in something or other, I don't doubt. Anyway, ramble on chaps. (Why are there no crank chapesses?) Brian Chandler http://imaginatorium.org
From: Virgil on 10 Oct 2006 02:41 In article <1160461892.665188.315170(a)i42g2000cwa.googlegroups.com>, imaginatorium(a)despammed.com wrote: > Dik T. Winter wrote: > > In article <1160377400.288823.275240(a)c28g2000cwb.googlegroups.com> > > cbrown(a)cbrownsystems.com writes: > > > Dik T. Winter wrote: > > ... > > > > The balls in vase problem suffers because the problem is not > > > > well-defined. > > > > Most people in the discussion assume some implicit definitions, well > > > > that > > > > does not work as other people assume other definitions. How do you > > > > *define* the number of balls at noon? > > > > > > I disagree. > > > > Do you? > > > > > This is no more difficult than asking "how do you *define* the number > > > of balls at pi/10 seconds before midnight?" > > > > And you start giving a definition. I do not say that it is *difficult* to > > define. But in my opinion more than one definition is possible. > > I agree with Chas - this simply isn't anything to be _defined_. Of > course the problem as stated is in real-worldy-looking terms, as though > this were plausibly an experiment one could do (can you say "intuition > pump"), and it is necessary to _define_ how the statement is to be > interpreted mathematically. This seems to be fairly simple (forgive me > if I use slightly the 'wrong' formula, corresponding to my "sliver" > example, but it makes no real difference): > > For every natural number n there is a ball, and this ball has two > states along a timeline (value represented by t): > > When -2/n <= t <= -1/n state is IN > When t < -2/n or t > -1/n state is OUT I think the timelines are a bit more complicated than this in the originally stated problem, but you have nailed the essence, that for each n there is an interval strictly prior to t = 0 during which the state is IN, and at all other times the state is OUT. > > If the question is: For how many of these balls is the status IN at > t=0, the answer is completely clear: none at all. No further > "definition" is needed, or even possible. > > The cranks universally proceed from some intuitions about the real > world that - essentially - there are no discontinuous functions in > physics. If there are no discontinuous functions, then it follows that > you can "swap limits" - and in particular that if balls(t) is a > function representing the number of balls in state IN at time t, then > lim t->0 (balls(t)) = balls(0). If there _are_ discontinous functions, > this clearly does not follow. > > I do find it hilarious that one of the cranks proudly displays a graph > of y = 1/x on his website, and seems to believe this tells us "the > value of 1/0". Well, a little education would be no idleness in > something or other, I don't doubt. > > Anyway, ramble on chaps. (Why are there no crank chapesses?) > > Brian Chandler > http://imaginatorium.org
From: William Hughes on 10 Oct 2006 08:21 imaginatorium(a)despammed.com wrote: > Dik T. Winter wrote: > > In article <1160377400.288823.275240(a)c28g2000cwb.googlegroups.com> cbrown(a)cbrownsystems.com writes: > > > Dik T. Winter wrote: > > ... > > > > The balls in vase problem suffers because the problem is not well-defined. > > > > Most people in the discussion assume some implicit definitions, well that > > > > does not work as other people assume other definitions. How do you > > > > *define* the number of balls at noon? > > > > > > I disagree. > > > > Do you? > > > > > This is no more difficult than asking "how do you *define* the number > > > of balls at pi/10 seconds before midnight?" > > > > And you start giving a definition. I do not say that it is *difficult* to > > define. But in my opinion more than one definition is possible. > > I agree with Chas - this simply isn't anything to be _defined_. Of > course the problem as stated is in real-worldy-looking terms, as though > this were plausibly an experiment one could do (can you say "intuition > pump"), and it is necessary to _define_ how the statement is to be > interpreted mathematically. This seems to be fairly simple ... I straddle the fence a bit. I agree with Dik Winter that a defintion of the set of balls in the vase at noon is needed, however, I agree with you that there is one very obvious definition of this set (the balls that are added before noon and not removed before noon), nor can I see any other sensible definitions (clearly one can create arbitrary definitions, e.g. the set of balls at noon consists of the ball 47, but these are not very interesting). The problem is that the question asks not i. What is the set of balls in the vase at noon. but ii. How many balls are in the vase at noon. One obvious approach to answering ii is -define the set of balls in the vase at noon -determine what this set is -determine the cardinality of the set However, there is an alternate approach. Define the number of balls in the vase at noon directly. Here we have two intuitive approaches a: the number of balls in the vase at noon is the cardinality of the set of balls in the vase at noon. b: the number of balls in the vase at noon is the limit of the number of balls in the vase as t approaches noon a and b are of course contradictory if we use the "natural" definiton of the set of balls in the vase at noon. Those who want to insist on definiton b have three choices I Infinite sets are contradictory (however this contradiction depends on b being true, and there is no proof within set theory that b is true, so this "contradiction" is based on intuition) II The number of balls in the vase at noon has nothing to do with the cardinality of the set of balls in the vase at noon (this option is not attractive, and no one chooses it) III Use some other definition for the set of balls in the vase than the "natural" definition. (Although at times the necessity of doing this is admitted, the intuitive appeal of the "natural" definiton is so strong that it has not been done. At best a set of magic balls, consisting of balls labelled with "undeterminable integers", or "infinite integers" is posited. The properties of these balls are not specified, indeed it may be said that the properties cannot be specified, hence it cannot be known that these balls are not in the vase. Since some balls must be in the vase, these balls must be in the vase (why these balls and not some other balls such as the "hyper-indeterminable" balls is not made clear).) - William Hughes
From: Han.deBruijn on 10 Oct 2006 12:12
David R Tribble schreef: > Mueckenheim wrote: > >> But you cannot derive that the vase is not empty at noon from the > >> observation that its contents cannot decrease? > > Han de Bruijn wrote: > > A picture says more than a thousand words. [Doesn't] it? > > > > http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > > I notice that there is no Y point at the rightmost X at "noon". True. That symbolizes the fact that there is no noon. It's also a fact that you cannot do something else with infinity than clipping it against the window, graphically speaking. Han de Bruijn |