Cantor Confusion
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From: William Hughes on 5 Oct 2006 07:14 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. 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: mueckenh on 5 Oct 2006 07:23 Hi, Dik, I would like to publish our result to the mathematicians of this group in order to show what they really are believing if they believe in set theory. There is an infinite sequence S of units, denoted by S = III... This sequence is covered up to any position n (included) by the finite sequences I II III .... But it is impossible to cover every position of S. So: S is covered up to every position, but it is not possible to cover every position. A further discussion is not useful, but the statement should be known. Regards, WM
From: William Hughes on 5 Oct 2006 07:23 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". > 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. - William Hughes
From: Albrecht on 5 Oct 2006 07:38 Virgil schrieb: > In article <1160033333.428361.122020(a)h48g2000cwc.googlegroups.com>, > "Albrecht" <albstorz(a)gmx.de> wrote: > > > MoeBlee schrieb: > > > > > Albrecht wrote: > > > > MoeBlee schrieb: > > > > > > > > > Albrecht wrote: > > > > > > > > > > > I will try to explain my claim again: There is only a[n] > > > > > > [anti-]diagonal number > > > > > > which is proveable not in the list if there is a real number which is > > > > > > build up out of all the real numbers in the list. But for an infinite > > > > > > list you can't end the diagonal number. You may have a sequence which > > > > > > converge. But you never have a limit. > > > > > > > > > > Then you're not talking about the reals. > > > > > > > > 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. > > There are a lot more lines in R2 for which neither neither coordinate is > constant, There are only two of infinitely many directions for which the > lines are not skew. To whom do you talk? R2 was not my theme. I'm ready with R. Maybe you want adress your information to Moeblee? > > > > > > > > > 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. > > But 99% is not good enough, in mathematics. No, todays math needs 1000%. As it needs infinite^2 and more objects. Best regards Albrecht S. Storz
From: georgie on 5 Oct 2006 08:37
Virgil wrote: > If "georgie" doesn't like my posts, why doesn't he/she/it killfile them? It's informative to see who is worthless. |