Cantor Confusion
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From: William Hughes on 11 Oct 2006 06:45 mueckenh(a)rz.fh-augsburg.de wrote: [...] > There is no largest natural! There is a finite set of > arbitrarily large naturals. The size of the numbers is unbounded. > I can only conclude you have knocked youself out. - William Hughes
From: mueckenh on 11 Oct 2006 06:47 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > My conclusion is: > > > > Either > > > > (S is covered up to every position <==> S is completely covered by at > > > > least one element of the infinite set of finite unary numbers > > > > > > Straight quatifier dyslexia. The fact that "for every x there exists > > > a y such that" does not imply "there exists a y such that for every x" > > > > A nonsense argument. > > Hardly. For every integer x there exists an integer y such that > x+y = 0. This does not imply that there exists a y such that for > every x, x+y=0. So in general the implication does not hold. Uninteresting. > > You claim it holds in a specific case. Correct. > > >Your assertion is wrong in a linear set. > > Give an > > example where the linear set covers a number which is not covered by > > one member of the linear set. > > > > > [If I understand your definition of cover] This is not possible. A > number > cannot be covered by a set but only by a member of a set. Correct. But it is asserted that 0.111... is not covered by a member of the list 0.1 0.11 0.111 .... but that it is covered by the whole list. You are correct. That is impossible. > But this is a red herring. What we want is > > Given two sets, a linear set A and another set B, such > that for every x in B there is a y in A such that y covers x. > Then there is a y in A such that for every x in B, y > covers x. > > It is easy to show that this can be false, if and only if > A is a linear set with no largest element. I.e. it is > not true. What is implied? Not my assertion is false but the assertion s false that there are finished infinte sets. This has become obvious by the linear example of unary numbers. It had not yet been obvious because all matematician relied on such non-linear examples like those given above. Regards, WM
From: mueckenh on 11 Oct 2006 06:57 David R Tribble schrieb: > William Hughes schrieb: > >> Note, the question originally asked was very careful to > >> distinguish between the questions " Will the whole autobiography > >> be written?", and "Will certain pages of the autobiography > >> be written?, so my repharasing is accurate. > > > > MueckenH wrote: > > Yes, but the assertion of Fraenkel and Levy was: "but if he lived > > forever then no part of his biography would remain unwritten". That is > > wrong, because the major part remains unwritten. You see it by havin > > Tristram Shandy write only his firsts of January at unchanged speed. > > Which parts remain unwritten? Do you have a particular range of > days in mind? If t denotes the year then 364*t days remain unwritten. Which parts belong to that set depends on t. If you ask me what f(x) = x^2 gives, then you cannot expect an answer unless you specify x. Nevertheless I can confirm that for x =/= 0 f(x) is larger than 0. > > > > With potential infinity there is no contradiction. There it is > > meaningless to consider noon, i.e. to consider the completed set, i.e. > > to consider every ball. > > Which means that there must be some balls we can't consider, which > are left out of the "set of all balls", right? Right. So it is in fact. The next one after the last you have considered is always left out. > > > > If, however, the whole set of N is considered as actually existing, > > then there is a contradiction, because then the union of all natural > > numbers is a fixed set which does not leave room for further numbers. > > Once you have all of the naturals in the set, what "further" naturals > are there? Did you accidentally leave some out? In this example you cannot meaningfully speak of having all naturals. All are out of the vase and some more remain inside. There is no set of all naturals. > > > > Then "each" is contradictive because we know that there is a set of > > numbers which is not removed and which has a larger (precisely: not a > > smaller) cardinal number than the set of numbers removed. > > Which ones are not removed? This question implies the existence of a completed set. Therefore it is meaningless. > > > William Hughes schrieb: > >> If it gives you a warm fuzzy to say that > >> "Every ball will be removed at some time before noon", > > > > MueckenH wrote: > > No. To say that every ball will be removed, is wrong, because there is > > not every ball. > > Where are all those missing balls? Are they in some other set or vase? They are not anywhere. Regards, WM
From: mueckenh on 11 Oct 2006 07:02 David R Tribble schrieb: > William Hughes schrieb: > >> "For any natural N, the ball numbered N will be removed from > >> the vase before noon" > > > > MueckenH wrote: > > There is not "any natural" but only those which we can define. There is > > a largest natural which ever will be defined. Hence mathematics in the > > universe and in eternity has to do with only a very small sequence of > > naturals. > > Let's give that largest natural a name: M. So the largest natural > which will ever be defined (presumably by the number elves that > define all the numbers that we humans use) is M. Then, by your > definition, M+1 cannot be a number. False. Then M+1 would be M. Therefore M is not a fixed number but depends on what has been done. Anyway, it exists, because no one can name a larger natural than he can name. > We can't even talk about it or > hypothesize anything about it, because it's just too large to be a > number, Then it s no a number. Hence it is not a natural. > and the universe won't exist long enough for anyone to > define it (even though you said that the numbers exist in eternity - > I guess that's a short eternity, not the long kind?). > > I wonder, though, is M odd or even? Of course, I can't speculate if > M+1 is odd or even, though, because it's been defined that I can't. You can speak of it. It is a natural. But you will not speak of a larger one. > > > > Writing 1,2,3,... is but cheating > > Which of your rules does it break? The assumption of the complete set is inconsistent as we have seen by discussion the vase. Regards, WM
From: William Hughes on 11 Oct 2006 07:15
Albrecht wrote: > William Hughes schrieb: > > > Albrecht wrote: > > > William Hughes wrote: > > > > Albrecht wrote: > > > > > William Hughes wrote: > > > > > > Albrecht wrote: > > > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > Albrecht wrote: > > > > > > > > > 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. > > > > > > > > > > > > > > > > Don't you find it interesting that of all the places you looked, > > > > > > > > the only place where anyone disagreed with the diagonal > > > > > > > > argument was the newsgroups? > > > > > > > > > > > > > > That's really untrue. I had read several books and papers of academics > > > > > > > (who do not post in this or the german math newsgroups) in which they > > > > > > > formulate (very cautious) criticism about ZF, axiomatic set theory or > > > > > > > especially the axiom of infinity. > > > > > > > > > > > > Did any of them disagree with the diagonal argument? > > > > > > > > > > > > > > > I think, the answer to this question don't enlight the whole problem we > > > > > discuss. > > > > > > > > In other words, no. > > > > > > > > > > > > Now read the rest of the post. > > > > > > > > > > > > > > Since you choose to don't read the rest of my post, why should I do > > > otherwise? > > > > Two reasons. > > > > You were trying to change the subject, I was not. > > I posted first. > > > > > Which subject? That the axiomatic method dominate modern math cause > Cantor "invented" the axiom of infinity and the diagonal argument? The > first idea is an arbitrary assumption, the second one bases on > arbitrary assumptiones. Both ideas belong together. > The point is that while the diagonal argument is based on what you term "arbitrary assumptions", it follows logically from those assumptions. Despite this, it is common for people on this newsgroup to argue "I don't like the result of the diagonal argument therefore it cannot follow logically" and try to attack the logic of the argument. If you want to talk about the "arbitrary assumptions" then you are talking about axioms. You may claim that you do not use axioms, but you must (explicitely or implicitely) make assumptions. - William Hughes |