Cantor Confusion
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From: Randy Poe on 6 Oct 2006 06:21 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > 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? > > In this case omega is enforced. Even if I knew what you meant by "enforcing omega", it wouldn't apply here. This person will never achieve an infinite age by adding one day at a time. - Randy
From: mueckenh on 6 Oct 2006 06:22 Tonico schrieb: > > > 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. You should know that Cantor designed his theory in order to describe real life, physics, chemistry, economy, ... > 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. > 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. > 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. > 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. > ************************************************************************* > > > > 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. In sums like 1/2 + 1/4 + 1/8 + ... noon does exist. > > > > 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. If you consider it as existing, then we should have access to it and Mr. X too. Or what is your existence else? Something like the well-order of any set? Existing but never accessible? Regards, WM
From: mueckenh on 6 Oct 2006 06:37 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. Regards, WM
From: Albrecht on 6 Oct 2006 06:42 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. We prove that every convergent > sequence of rationals converges to a unique real number, which is the > limit of the sequence. If it converges, then it has a limit. Who had proved this for arbitrary infinite sequences of decimal digits? > Thus every > countable decimal expansion represents a real number. In particular, > the anti-diagonal of any given countable sequence of denumerable > decimal expansions is a countable decimal expansion and thus represents > a real number. And we prove it does not represent any real number in > the given countable sequence of decimal expansions. > > Argue that you don't like the axioms of set theory, if you like. Argue > that you don't ascribe to the principles of reasoning codified by first > order logic, if you like. But it is not in any way rationally arguable > that the uncountability of the reals is not a theorem of the stated > axioms. And more simply, there is no theory that you have proposed in > which there exists a function from the counting numbers onto the > carrier set of a complete ordered field. > > > Cantor argues that you must not have the limit. > > We must not have the limit of what? And where does Cantor argue this? The limit of the antidiagonal. > Morevover, no matter what Cantor did or did not write, we've moved on > to axiomatic theories that are not at all beholden to Cantor. > > > But with the same idea > > you can e.g. construct a kind of an [anti-]diagonal (natural) number of any > > list of natural numbers. > > So what? If it's a denumerable list of natural numbers, then there is > no anti-diagonal that is a finite sequence or a denumerable that is all > 0's after some position, hence no such anti-diagonal represents a > natural number, since natural numbers are represented only by finite > sequences or by sequences that are all 0's after some position. > > > In > > consequence the set of the natural numbers is uncountable. > > That is incorrect. See above paragraph. > > > You can proof a lot of strange things with the idea of Cantor. > > Many of the ideas originate with Cantor, but the axiomatizations are > not beholden to Cantor. > > > But > > these things are not very useful. > > They axiomatize ordinary calculus used for science and engineering. > Moreover, mathematical logic and set theory were and are very useful in > the development of the digital computer, as you are using such a > digital computer to declare that these are ideas are not very useful. > Without axiom of infinity no computers? Computer compute. Befor computers compute, men compute. We have to through away ZF and then compute again by hand? How bad! :-) Best regards Albrecht S. Storz
From: mueckenh on 6 Oct 2006 06:43
Virgil schrieb: > > Is there a day in X's life, such that less than 50,000 years remain to > > be written? > > At any given time there are years worth of days waiting to be written > about, but each day will have its day. And at each day there will be further years. > > > 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? > Why do you think it is meaningful to assert otherwise? > A process which never ends can do an unlimited amount. And can leave an unlimited amount. Switch to Tristram Shandy who is more instructive. 99% of his life remain unwritten forever. Regards, WM |