Cantor Confusion
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From: stephen on 17 Oct 2006 09:21 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> >>>stephen(a)nomail.com wrote: >> >>>>You have not answered the question about how one determines >>>>if a thing in the real world is true. I can guess you >>>>will say something about measurements, but how does one >>>>know that your measurements are "true", or that they truly >>>>correspond to "a thing in the real world", and so on. >>>>It is a big ugly kettle of philosophical fish. >> >>>Not only philosophical fish. Also religious fish. And political fish. >>>And scientific fish. Actually everyday's life fish. You are right! >> >>>>I agree that it is sensible to assume that the Universe >>>>is consistent, but given how strange and unintuitive >>>>the Universe can be, who knows. >> >>>I think we agree on the above. But it doesn't mean that we cannot answer >>>_part_ of the question: do INFINITIES exist or not. Are they true or are >>>they false? And IMO _that_ can be decided _now_, without rocket science. >> >> No, I do not see how we can decide that. We cannot observe >> infinities, but that does not mean they do not exist. Unless >> of course that is some axiom of yours, but again, you cannot >> know that that axiom is "true". >> >> There are people who are convinced that the Universe is truly >> eternal, that is always has been, and always will be. I suppose >> their arguments have convinced them that an infinity does exist. >> What irrefutable evidence do you have that the Universe has >> not always existed? > The scientific attitude: thy shall not believe what thy cannot measure. > Han de Bruijn That is an axiom, aka an assumption, and it is not really believed by scientists. For example, scientists believe the photon truly has zero mass. Of course you cannot measure that, but that does not change the fact it is a "belief" of the standard theory. Stephen
From: stephen on 17 Oct 2006 09:27 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Anyway, how NOT EXACTLY IS the world? Is there an aether? >> Do photons have mass? Share your secrets. > http://hdebruijn.soo.dto.tudelft.nl/QED/ > Han de Bruijn No mention of aether or photon mass. Perhaps you do not now how NOT EXACTLY IS the world afterall. Stephen
From: Dik T. Winter on 17 Oct 2006 11:07 In article <1161079802.120515.175530(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > The inconsistency is that > 1) For the balls inserted until noon, you can find the result: It is > the set N. > 2) For the balls removed until noon, you can find the result: It is the > set N. > 3) For the balls remaining at noon, the same arguments of continuity > which lead to (1) and (2) cannot apply. There are quite a few obvious reasons. (1) 1) is not because of continuity (2) 2) is not because of continuity (3) no continuity reasoning can lead to the result that the balls remaining at noon is the set N. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: imaginatorium on 17 Oct 2006 11:20 mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Randy Poe schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Randy Poe schrieb: > > > > > > > > > > > Han de Bruijn wrote: > > > > > > > > I merely note that there is no requirement in the problem that > > > > > > > > the limit be the value at noon. > > > > > > > > > > > > > > The limit at noon - iff it existed - would be the value at noon. > > > > > > > > > > > > Wrong. That is a flat out incorrect statement showing a > > > > > > fundamental misunderstanding about what limits mean. > > > > > > > > > > > > A CONTINUOUS function at x0 has the property that the > > > > > > limit of f(x) as x->x0 is f(x0). But not all functions are > > > > > > continuous. > > > > > > > > > > And you are in charge of determining which functions are continuous and > > > > > which are not? > > > > > > > > Where do you get this stuff from? > > > > > > > > How do you translate a statement that some functions are not > > > > continuous into "I am in charge of determining if some functions > > > > are continuous"? > > > > > > Because it seems a bit mysterious, how you know or can define that the > > > function of balls in the vase was not continuous. Or, may be, because > > > you will accept that the function is continuous if the balls are taken > > > out in the sequence 1, 11, 21, .... > > > > > > > > No, I am not in charge. Non-continuous functions are non-continuous > > > > now and forever. They were non-continuous before I existed, they will > > > > remain non-continuous after I'm gone. > > > > > > > > The number of balls in the vase is such a function. > > > > > > How do you acquire that knowledge? > > > > By the application of a little elementary mathematical knowledge, I > > should think. > > > > (1) For any positive value nu (integer, but actually a real will do > > too), for a sufficiently small value of tau, the number of balls in the > > vase at time noon-tau is greater than nu. This can be derived > > tediously, but obviously, from the set of step functions, one for each > > natural n, representing the state of the nth ball. Therefore the limit > > to the number of balls as time approaches noon from the "left" > > diverges. > > > > (2) For the value t=noon, there does not exist any n for which the step > > function representing the state of ball n has the value IN. Therefore > > the set of balls with the corresponding value IN is empty, and the > > number of balls is zero. > > > > Therefore the limit as t->noon does not equal the value at noon. This > > simply follows from the statements in the problem. > > But the function of balls/numbers removed from the vase is a > continuously (stepwise) increasing one, containing all natural numbers > at noon? Uh, yes, unless I mysteriously misunderstand you... If takenout() is a function from time to the power set of the integers (i.e. it maps to a set of integers) then each natural number m is included in the set that takenout() maps to from time = -1/m. So by time zero, all natural numbers are included. Was there a question with that? Brian Chandler http://imaginatorium.org > > Regards, WM
From: Randy Poe on 17 Oct 2006 11:32
Han de Bruijn wrote: > stephen(a)nomail.com wrote: > > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > >>stephen(a)nomail.com wrote: > > > >>>You have not answered the question about how one determines > >>>if a thing in the real world is true. I can guess you > >>>will say something about measurements, but how does one > >>>know that your measurements are "true", or that they truly > >>>correspond to "a thing in the real world", and so on. > >>>It is a big ugly kettle of philosophical fish. > > > >>Not only philosophical fish. Also religious fish. And political fish. > >>And scientific fish. Actually everyday's life fish. You are right! > > > >>>I agree that it is sensible to assume that the Universe > >>>is consistent, but given how strange and unintuitive > >>>the Universe can be, who knows. > > > >>I think we agree on the above. But it doesn't mean that we cannot answer > >>_part_ of the question: do INFINITIES exist or not. Are they true or are > >>they false? And IMO _that_ can be decided _now_, without rocket science. > > > > No, I do not see how we can decide that. We cannot observe > > infinities, but that does not mean they do not exist. Unless > > of course that is some axiom of yours, but again, you cannot > > know that that axiom is "true". > > > > There are people who are convinced that the Universe is truly > > eternal, that is always has been, and always will be. I suppose > > their arguments have convinced them that an infinity does exist. > > What irrefutable evidence do you have that the Universe has > > not always existed? > > The scientific attitude: thy shall not believe what thy cannot measure. You are confused. This is an attitude about the physical world. The scientific method is silent about the mathematical world. - Randy |