Cantor Confusion
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From: Randy Poe on 5 Oct 2006 10:27 mueckenh(a)rz.fh-augsburg.de wrote: > Tonico schrieb: > > > Han de Bruijn wrote: > > > Tonico wrote: > > > > > > > Han de Bruijn ha escrito: > > > > > > > >>Wishful thinking on the part of William Hughes. A simple Google search > > > >>will reveal that the army of 'sci.math' dissidents is steadily growing. > > > > > > > > ************************************************** > > > > Zaas! Dissidents...from sci.math???? Didn't know somebody already > > > > formed a political or social or economical or > > > > whatever-that-isn't-science group called sci.math, and that it already > > > > has its dissidents! Perhaps Han means people that insist in talking > > > > about mathematics with some mathematicians from a non-mathematical > > > > point of view and without knowing mathematics? People that attack > > > > mathematicians and even mathematics (go figure!) when someone dares to > > > > point out some mathematical mistake in some nonsense that THEY say is > > > > correct IN SPITE of evidence in contrary? > > > > > > One cannot speak of correct with a mathematics that is a non-discipline > > > and gives non-discplinary answers to ill-posed questions like this one: > > > > > > http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#bv > > > > > > Zero balls at noon? Carl Friedrich Gauss would have turned in his grave. > > > > > > Han de Bruijn > > > > ************************************************** > > For what we know, I think dead people don't turn in their grave, or > > anywhere else for that matter. I agree though that the balls-vase-noon > > is a question ill-posed but with possibilities to be pretty interesting > > and deep into understanding some aspects of infinity and stuff. > > I'd rather pose the next thinker. > > Supose X is a person that never dies, and when he's 50,000 years old > > he begins writing his autobiography > > > That is not far from the story of Tristram Shandy. > > > in a rather peculiar way: every day > > after his 50,000-th birthday he writes down one day of his life, > > beginning with his first day of life. The question is: does X write > > down ALL the days of his life in this autobiography? Pay attention: I > > am not asking whether there will ever be a book containing all the days > > of X's life, but rather whether there will exist some pages written by > > X describing the events of some given day of his life, for EVERY > > singled out day of X's life...? > > Why do you think this question be more important or interesting than > the other? > But the answer is easy: He never writes down the next day, where "next" > is a number like "omega". So you think that counting days one at a time beginning with the 50,000 year, one of those steps will bring you to "a number like omega"? In other words, the process of counting finite natural numbers comes to an end? - Randy
From: mueckenh on 5 Oct 2006 12:34 William Hughes schrieb: > > Why do you think this question be more important or interesting than > > the other? > > But the answer is easy: He never writes down the next day, where "next" > > is a number like "omega". > Since the question never mentions a "next day", you answer does > not address the question. Try again. > > Will there exist some pages written by > X describing the events of some given day of his life, for EVERY > singled out day of X's life? > > > Note that "omega" is not a day of X's life, so an answer about > "omega" will not answer the question. Since your question does not address my question (the second sentence was a joke) it remains your turn to try again: Is there a day in X's life, such that less than 50,000 years remain to be written? Why do you think this question be less important than yours? And if not, why do you think that it is meaningful to assert that X could write about all his days? Regards, WM
From: MoeBlee on 5 Oct 2006 13:29 Albrecht wrote: > MoeBlee schrieb: > > > The points on a straight line are representations of the reals. Do you > > > think so? > > > > No, I don't think they are "representations" of real numbers. In > > analytic geometry, in, for example, the plane R2, they are ordered > > pairs of real numbers. > > Now let one of the numbers of the pairs constant and forget it and we > meet. I don't understand what you're saying. > > > With Cantor you have to think that some points on the > > > straight line (to be exact: the absolute majority of them) are not > > > reachable by construction. > > > Should this be a consequence of an useful mathematics? > > > > Give me your axiomatization that is rich enough for ordinary calculus, > > and with a mathematical definition of 'constructible', and such that > > every real number is constructible. Then we'll talk about that. > > Do we really need such an axiomatisation? 99% of mathematics was done > without it. And was done right. I am interested in formally axiomatized theories, mainly since these allow complete objectivity as to whether a purported proof is indeed a proof. Indeed, that this is possible even if only in princple is part of my own notion of mathematical proof "done right". Moreover, I am not personally satisfied by theories that I can't see how to formalize (at least if only in principle to formalize). I may see value in certain unformalized theories, but I still would have it as part of my personal program to learn how such unformalized theories would be formalized. That includes historical mathematics. If there are historical arguments for mathematical propositions such that the arguments cannot be formalized even if only in principle, then I am at least skeptical of those arguments. But there is a more present problem with your argument about 99% of mathematics. In our discussion, I asked about your mathematics. Your mathematics is not necessarily the same as the 99% of mathematics you mentioned. So that leads back to my point, even if I were to drop the requirement of formal axiomatization (which, ultimately, I would not drop): What mathematics do you offer that defines 'constructible' and has only constructible real numbers providing for ordinary calculus? I can't presume that your mathematics is the same as the 99% of mathematics that you mention. And, as I mentioned, even if you were to give an informal mathematics that answers my question, I would not, for the reasons I mentioned, be convinced by your mathematics if I were not able to see how to formalize it, even if only in principle. Moreover, as to caclulus, what is this 99% you have in mind? Aren't there changing ideas over history, from Newton and Leibniz through Cauchy, Weierstrass, Dedekind, Whitehead and Russell, to fully formalized set theory (and with alternatives in Brouwer through later intutionists)? At what point do you consider your 99% of mathematics to have culminated? > > I don't know what you mean by "endlessly converging values". What is > > the difference between converging and "endlessly converging"? > Surely my explanation was not very well done. But I feel that you don't > want more explanations about my standpoint anyway. > I don't controvert the axiomatic methode anymore. But I claim that it > isn't the only and the important one in math. In teaching and in the > mind of the people the axiomatic method appears to be the only right > way to do math. That's not correct. > The nondenumerable infinity of the reals is not the only one truth. > Nobody is wrong who claims only one kind of infinity, the one we only > can know: the endless infinity. I don't claim that axiomatics exhaust all of mathematical thinking. Mathematics includes discovery, heuristics, imagination, and other kinds of thinking. However, the primary subject matter of public discourse on mathematics is that of stating theories, proving theorems, and stating conjectures about what are or are not theorems of certain theories. And even as to discussion of more ruminative mathematical thinking, often, at a certain point, it is needed to ask for some context of a formal theory lest the discussion descend into a cacophonoy of arguments over utterly subjective mental imagery. MoeBlee
From: Tonico on 5 Oct 2006 14:17 mueckenh(a)rz.fh-augsburg.de wrote: > Tonico schrieb: ......................................... One cannot speak of correct with a mathematics that is a non-discipline > > > and gives non-discplinary answers to ill-posed questions like this one: > > > > > > http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#bv > > > > > > Zero balls at noon? Carl Friedrich Gauss would have turned in his grave. > > > > > > Han de Bruijn > > > > ************************************************** > > For what we know, I think dead people don't turn in their grave, or > > anywhere else for that matter. I agree though that the balls-vase-noon > > is a question ill-posed but with possibilities to be pretty interesting > > and deep into understanding some aspects of infinity and stuff. > > I'd rather pose the next thinker. > > Supose X is a person that never dies, and when he's 50,000 years old > > he begins writing his autobiography > > > That is not far from the story of Tristram Shandy. > > > in a rather peculiar way: every day > > after his 50,000-th birthday he writes down one day of his life, > > beginning with his first day of life. The question is: does X write > > down ALL the days of his life in this autobiography? Pay attention: I > > am not asking whether there will ever be a book containing all the days > > of X's life, but rather whether there will exist some pages written by > > X describing the events of some given day of his life, for EVERY > > singled out day of X's life...? > > Why do you think this question be more important or interesting than > the other? > But the answer is easy: He never writes down the next day, where "next" > is a number like "omega". ************************************************************************* Whoever talked bout "interesting" or "important"? I just pointed out that the vase-balls question was an ill-posed one since it mixed maths and real life in an improper way, imfho. Of course, many thinkers are designed precisely that way: as language tricks to confuse others. About omega: are you talking of the ordinal of the natural numbers in their natural ordering? What does that have to do with this? What is "a number like omega", anyway? Do you mean a limit ordinal? Again: what that has to do with the thinker I wrote? How does one eternal being go about being "omega" years old? And beside and beyond all this: as you most certainly know from basic set theory, omega and other limit ordinals are that: limit ordinal, precisely because they are not consecutive to any other ordinal, so "the omega day" being "the next" day is a rather pantagruelic stretch even in an imaginative story like the one I posted... ************************************************************************* > > Of course, the above story is very similar to the one about the hotel > > in space witrh infinite rooms in it, etc. > > Regards > > Tonio > > Ps. BTW abiut the balls-vase matter, I think the best answer is the > > one that said that noon, as we know it, is never reached according to > > the spirit of the question. > > And the same answer is appropriate if considering the whole set N: It > does not exist. ************************************************************************** You can consider the set N as not existing: I and number 7 don't really care. Regards Tonio > Regards, WM
From: Virgil on 5 Oct 2006 14:20
In article <1160046437.594941.91180(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Tonico schrieb: > > > Han de Bruijn wrote: > > > Tonico wrote: > > > > > > > Han de Bruijn ha escrito: > > > > > > > >>Wishful thinking on the part of William Hughes. A simple Google search > > > >>will reveal that the army of 'sci.math' dissidents is steadily growing. > > > > > > > > ************************************************** > > > > Zaas! Dissidents...from sci.math???? Didn't know somebody already > > > > formed a political or social or economical or > > > > whatever-that-isn't-science group called sci.math, and that it already > > > > has its dissidents! Perhaps Han means people that insist in talking > > > > about mathematics with some mathematicians from a non-mathematical > > > > point of view and without knowing mathematics? People that attack > > > > mathematicians and even mathematics (go figure!) when someone dares to > > > > point out some mathematical mistake in some nonsense that THEY say is > > > > correct IN SPITE of evidence in contrary? > > > > > > One cannot speak of correct with a mathematics that is a non-discipline > > > and gives non-discplinary answers to ill-posed questions like this one: > > > > > > http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#bv > > > > > > Zero balls at noon? Carl Friedrich Gauss would have turned in his grave. > > > > > > Han de Bruijn > > > > ************************************************** > > For what we know, I think dead people don't turn in their grave, or > > anywhere else for that matter. I agree though that the balls-vase-noon > > is a question ill-posed but with possibilities to be pretty interesting > > and deep into understanding some aspects of infinity and stuff. > > I'd rather pose the next thinker. > > Supose X is a person that never dies, and when he's 50,000 years old > > he begins writing his autobiography > > > That is not far from the story of Tristram Shandy. > > > in a rather peculiar way: every day > > after his 50,000-th birthday he writes down one day of his life, > > beginning with his first day of life. The question is: does X write > > down ALL the days of his life in this autobiography? Pay attention: I > > am not asking whether there will ever be a book containing all the days > > of X's life, but rather whether there will exist some pages written by > > X describing the events of some given day of his life, for EVERY > > singled out day of X's life...? > > Why do you think this question be more important or interesting than > the other? > But the answer is easy: He never writes down the next day, where "next" > is a number like "omega". By your argument, he never lives that day either, so he has never need write it down. |