Cantor Confusion
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From: Dik T. Winter on 9 Oct 2006 19:29 In article <virgil-9E9CC6.02103209102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> 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? .... > How about the following model: And you also start with definitions, or a model. I did *not* state that it was difficult to define, or to make a model. But without such a definition or model we are in limbo. I think other (consistent) definitions or models are possible, giving a different outcome. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 9 Oct 2006 19:27 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: William Hughes on 9 Oct 2006 19:36 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > My conclusion is: > > > Either > > > (S is covered up to every position <==> S is completely covered by at > > > least one element of the infinite set of finite unary numbers > > > > Straight quatifier dyslexia. The fact that "for every x there exists > > a y such that" does not imply "there exists a y such that for every x" > > A nonsense argument. Hardly. For every integer x there exists an integer y such that x+y = 0. This does not imply that there exists a y such that for every x, x+y=0. So in general the implication does not hold. You claim it holds in a specific case. >Your assertion is wrong in a linear set. > Give an > example where the linear set covers a number which is not covered by > one member of the linear set. > [If I understand your definition of cover] This is not possible. A number cannot be covered by a set but only by a member of a set. But this is a red herring. What we want is Given two sets, a linear set A and another set B, such that for every x in B there is a y in A such that y covers x. Then there is a y in A such that for every x in B, y covers x. It is easy to show that this can be false, if and only if A is a linear set with no largest element. I.e. it is not true. - William Hughes
From: Virgil on 9 Oct 2006 19:37 In article <J6w6LC.9rL(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: > In article <virgil-9E9CC6.02103209102006(a)comcast.dca.giganews.com> Virgil > <virgil(a)comcast.net> 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? > ... > > How about the following model: > > And you also start with definitions, or a model. I did *not* state that > it was difficult to define, or to make a model. But without such a > definition or model we are in limbo. I think other (consistent) definitions > or models are possible, giving a different outcome. Can you suggest one? One that does not ignore the numbering on the balls as some others have tried to do.
From: David R Tribble on 9 Oct 2006 22:23
William Hughes schrieb: >> Note, the question originally asked was very careful to >> distinguish between the questions " Will the whole autobiography >> be written?", and "Will certain pages of the autobiography >> be written?, so my repharasing is accurate. > MueckenH wrote: > Yes, but the assertion of Fraenkel and Levy was: "but if he lived > forever then no part of his biography would remain unwritten". That is > wrong, because the major part remains unwritten. You see it by havin > Tristram Shandy write only his firsts of January at unchanged speed. Which parts remain unwritten? Do you have a particular range of days in mind? > With potential infinity there is no contradiction. There it is > meaningless to consider noon, i.e. to consider the completed set, i.e. > to consider every ball. Which means that there must be some balls we can't consider, which are left out of the "set of all balls", right? > If, however, the whole set of N is considered as actually existing, > then there is a contradiction, because then the union of all natural > numbers is a fixed set which does not leave room for further numbers. Once you have all of the naturals in the set, what "further" naturals are there? Did you accidentally leave some out? > Then "each" is contradictive because we know that there is a set of > numbers which is not removed and which has a larger (precisely: not a > smaller) cardinal number than the set of numbers removed. Which ones are not removed? William Hughes schrieb: >> If it gives you a warm fuzzy to say that >> "Every ball will be removed at some time before noon", > MueckenH wrote: > No. To say that every ball will be removed, is wrong, because there is > not every ball. Where are all those missing balls? Are they in some other set or vase? |