Cantor Confusion
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From: Albrecht on 6 Oct 2006 06:46 Virgil schrieb: > In article <1160048322.932236.254800(a)b28g2000cwb.googlegroups.com>, > "Albrecht" <albstorz(a)gmx.de> wrote: > > > Virgil schrieb: > > > > > In article <1160033333.428361.122020(a)h48g2000cwc.googlegroups.com>, > > > "Albrecht" <albstorz(a)gmx.de> wrote: > > > > > > > MoeBlee schrieb: > > > > > > 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. > > > > > > But 99% is not good enough, in mathematics. > > > > No, todays math needs 1000%. As it needs infinite^2 and more objects. > > Actually, it is only A.S.S. who asks for 1000%. Mathematicians would > readily accept 100%, but not 99%. > > I'm sure: mathematicians have not 1%. To be exact: They never have any % of math as Goedel had shown. Math is infinite. Best regards Albrecht S. Storz
From: mueckenh on 6 Oct 2006 06:49 William Hughes schrieb: > Note the question was very carefully posed so it was not > "Can X write about all his days?", but "Can X write about > every single day?". There is the answer 1) There is no day which will not be written. There is the answer 2) There is a day which will not be written namely the present day. Both answers contradict each other. Regards, WM
From: Albrecht on 6 Oct 2006 07:13 William Hughes schrieb: > Albrecht wrote: > > <...> > > > 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. > > > > > The problem is not that someone who believes > in your intuitive "endless infinity" (intuitive because it cannot > be put on a mathematical footing) is wrong. Oh yes, it is the problem. I came to these subject by reading a bunch of popular books about math. When I read the diagonal argument the third or fourth time I started to wonder. The textes were of differnt quality but all of them had a special sort of feeling. And all stated, that this proof is so elementary, easy and absolute right that nobody had anything to reflect or critizise about it. But I found in shortest time a lot of questions about the issue. Later I read professional works about set theorie and I found a similar feeling in the textes about the diagonal argument. And then I started to learn about the role of ZF in the teaching on universities and I had a lot of disputes about the matter in newsgroups. And I learnd that there are a lot of people who react irrational if someone questions about the issue. In my opinion, the problem is that many people belief, they understand the diagonal proof - but they don't. And that a lot of people belief, that the axiomatic method is the top of math, as Hilbert had declared. And that a lot of people belief that math without ZF is dubious. And that a lot of people seem to have a religious feeling if they rant about transfinite numbers and unreachable cardinal numbers and all this stuff. ZF introduces a nonsientific element in math which is not beneficial for the science, I'm really sure. Good brains waste their time with transfinite numbers and all of this stuff and with fixing the problems which appear by using ZF. Math is much more. That's the way I see the problem. Best regards Albrecht S. Storz > Lots of people > have the same sort of intuitive feeling about infinity. Indeed, > the whole point of the "number of balls at noon" problem > is that it is counterintuitive. [However, a problem with an > informal definition is that you never really know if someone > agrees with you or not]. The problem is saying that > the mathematical definition of infinity is wrong. > > Statements like: > > Mathematicians believe that two types of infinity which > are different are really the same. > > Believing in the existence of a complete infinite set > is wrong. > > There is no such thing as the cardinality of the > natural numbers. > > are the problem. They imply that the usual mathematical definition > of infinity is inconsistent or wrong. > > If what you want to say is "My definition of infinity should be used > not yours", fine. But you cannot insist that it should be used > in mathematics. Your definition of infinity is mathematically useless, > so mathematicians will continue to ignore you. > > - William Hughes
From: Tonico on 6 Oct 2006 07:22 mueckenh(a)rz.fh-augsburg.de wrote: > Tonico schrieb: >> > ************************************************************************* > > 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. > > You should know that Cantor designed his theory in order to describe > real life, physics, chemistry, economy, ... ******************************************************************** I really didn't know that, and I very heavily suspect this isn't true, though I can't be 100% sure. What does foundations of maths in general, and trying to describe stuff about infinity from a mathematical point of view have in common with describing "real life", whatever that is?? What does math in general HAS TO HAVE in common with real life? In my opinion, nothing. Now if somebody ever finds some application to "real life" of some part of maths, as the cas has been MANY times and in some rather important occasions, good. This is the reason I can't think of Cantor developing his stuff because he wanted to describe whatever from "real life"... ****************************************************************************************************** > > 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? > > I am talking of the substitute for actual infinity, which Cantor > introduced. omega is not only the set N but also the cardinal number of > this set. ************************************************************************************** I wonder what do you call "actual infinity" to, and I'm afraid that this attitude of yours begins to ressemble a lot what sometimes little kids do: they cry, kick and make a fuss out of something just because life, or some part of it, happens not to be like they've decided it SHOULD be. Omega is NOT a set at all: it is the ordinal of the natural numbers set N with the usual ordering. Period. ****************************************************************************************** > > What does that have to do with this? What is "a > > number like omega", anyway? > > According to Cantor it is a whole number larger than any natural > number. It is my first aim to show that omega is not a number. ************************************************************************* ??!!!?!?! Did cantor EVER talk of a "whole number" (I supose you must mean an integer, positive one...and this already points towards the heavy doubt I had: you are not a mathematician, which of course doesn't automatically rule you out as a debater in these matters, but it does make some of your arguments highly suspicious and even dismissable from a mathematical point of view. Why? Because you simply doesn't know enough maths, apparently)?? No, he didn't...and again: omega is NOT any number at all, but an ordinal. No need for you to waste your better efforts in vain: omega has NEVER been a number. ******************************************************************************* > > 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? > > It has to do with the actual existence of infinity. It has to do with > the possibility to surpass this infinity by a larger one. **************************************************************************************** Since this is a maths list, I won't even get into a debate about what you can possibly mean by not-yet-defined notions as "actual existence" (as opossed to "just-kidding-folks!existence" or what?!), or what does "surpassing" infinities mean.... *********************************************************************************** > > 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... > > If you do not believe in actual infinity then we have no dissens. X > writes and writes and nothing happens. But for this sake we need no > symbol omega. And we know that never all day of his life are described > because always 50,000 remain to be written. ***************************************************************************************** This is maths: neitehr infinity nor differentiable functions, matrices or infinite (!!) abelian groups require or demand belief from someone. This is a wonderful, enjoyable and intellectual game called maths. Wanna play? Abide by its rules and/or propose rules that make sense mathematicallywise(??) to change it. Don't wanna play? Then don't be a mathematician (I think you already took this into account), and go do something else. As simple as that. About the "And we know that never all day of his life are described because always 50,000 remain to be written."[sic]: it'd be interesting if you could be more precise and point out who "we" in your sentence is. Regards Tonio Regards, WM
From: Tonico on 6 Oct 2006 07:31
mueckenh(a)rz.fh-augsburg.de wrote: > Tonico schrieb: > > > > 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? > > > ************************************************************************* > > If your question was "why is this more important..." , then I already > > answered this (since I posted the game I address this) in another post. > > About your question " Is there a day in X's life, such that less than > > 50,000 years remain to > > > be written?", the answer appears to be YES, since this guy begins writing his autobiography when his 50,000 years old exactly, so at the end of that day there remains to be written 50,000 years MINUS 1 day...and this relation will remain that way forever, as far as I can see. > > Oh, I overlooked that he needs only one day to write a whole year. Why > does he not continue the next day, so that he could catch up and in > fact get ready. > > > What I can't see is why is this important? This thinker is designed for people, in particular maths students, to think about the oddities and anti-intuitive shocks one usually gets when getting deep into set theory, >infinity and stuff. That's all. > > The original character is Tristram Shandy of a novel by L. Sterne who > needs one year to describe one of his days. (Let him start at birth.) > "If he lives forever, no day of his life remains unwritten" is the > assertion of single-eyed set theorists. More than 99% of his life > remain unwritten is the truth. Even if he lives forever. To teach that > to math students would be more important. ***************************************************************************** Nonsense. This is a mathematical question posed by mathematicians to mathematics students, so we abide by mathematics rules and not by our whims. Given the information of that Tristram Shandy story, the answer is pretty simple and even very easy to prove: EVERY single day of his life will be written down, even if it'd take 1 millions years to Tristram to write every day, and not merely one year. Does this shock your intuition? Good, that's how this is suposed to work: you can follow your intuition in maths, but be sure to apply LOGIC, AXIOMS and reason to confirm your intuition. Otherwise just go and be an engineer and say to all you know maths....**sigh**. You write "More than 99% of his life remain unwritten", and three words later YOU ALSO write "Even if he lives forever"...!!!! If you can't see the HUGE nonsense this is, EVEN from the standpoint of your rather bizarre opinions about infinity and stuff, then all is hopeless. Regards Tonio > Regards, WM |