Cantor Confusion
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From: Andy Smith on 5 Jan 2007 23:47 >> If you have an idealised bouncing ball that loses 1/2 >> its height on each bounce and takes 1/2 the time, >> how many bounces has it done in 2.00.. seconds? >> Hasn't it achieved an actual infinity of bounces as a >> finished thing? >It bounces infinitely many times. I don't know why you insist on adding >the meaningless phrase "as a finished thing". But if you number the >bounces sequentially, then each bounce gets a finite number. I am sure that you are right. To my uneducated mind such a thought experiment demonstrated the ball metaphorically counting up to omega, so that omega could be regarded as a natural number (and then of course there would be many others, which is exactly where I came in, with 2^omega real numbers and then a Zenoian counting sequence to enumerate them with a set of infinite integers (which cannot exist if omega is not a natural number). I will go and read a book and not waste your time further. Thanks
From: William Hughes on 6 Jan 2007 10:38 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > I do not assume that 0.111... exists. I only assume that > > > > given that 0.111...1 exists you can show that 0.111...11 exists. > > > > > > > That position is correct. It is called potential infinity. > > > > > > > > > > That is the same with the lines. Why should the diagonal exist actually > > > > > but the system of lines should not exist actually? (1/9 = 0.111...) > > > > > > > > Indeed. It is the same with lines. It is always possible to find > > > > another line. However, iii says nothing about whether the > > > > diagonal "actally exists" (or equivalently whether the > > > > system of lines "acutally exists"). > > > > > > But you always assert that the the complete diagonal exists, i.e., a > > > diagonal which could not be extended. Then you must also accept a line > > > system which cannot be extendend. > > > > No. At no time do I assume that the complete diagonal exists. > > > > > > > > > > > > > > > iv: Not all the elements of the diagonal exist > > > > > > > > > > Not all natural numbers in unary representation 0.1, 0.11, 0.111, ... > > > > > exist. If all elements of the diagonal exist (which are the last digits > > > > > of the unary numbers) then these unary numbers must exist too, as far > > > > > as I understand existence. > > > > > > > > Any discussion of iv is completely irrelevant. iv is neither needed > > > > nor used. > > > > > > Then drop it. > > > > I have never used it ("neither needed nor used"). > > > > > > > > You used to use ~iv: all the elements of the diagonal exist > > > > > > > Only to point out that I neither need nor use the > > assumption ~iv. > > > > > > The contradiction is: > > > > > > > > A: L_D must have the property that given any set of elements > > > > that exist in L_D one can always find another element of > > > > L_D > > > > [This follows immediately from i ii and iii] > > > > > > It is not different for the lines. > > > > > > > > B: L_D has a largest element. [this follows from the fact the > > > > L_D > > > > is a line] > > > > > > This follows from the assumption (iv) of complete existence of L_D, > > > which implies complete existence of the system of lines. If you drop > > > it, then the contradiction vanishes. > > > > No, it follows from the fact that L_D is a line. A line has a largest > > element. > > The diagonal has not a largest element, unless you refer to potential > infinity, where it always has a largest element but is not a fixed, > complete infinity. > > > L_D is not the system of lines. > > No assumption about the complete existence of the > > system of lines is needed or used. > > What do you mean to have proved? That L_D does not exist. Assume a line that contains any element that can be shown to be in the diagonal exists. Call this line L_D. A: L_D is a line, therefore L_D has a largest element. B: L_D contains any element that can be shown to exist in the diagonal, therefore L_D does not have a largest element. Contradiction. Therefore L_D does not exist. - William Hughes
From: mueckenh on 6 Jan 2007 13:10 Dik T. Winter schrieb: > > > Can you really believe that a thinking brain will accept your assertion > > that two absolutely identical systems of nodes and edges will supply > > different systems of paths, i.e., strings of nodes and edges? > > Yes, it entirely depends on how you find your paths. There are no paths to find. Paths do exist in the tree. > Consider a graph > consisting of three sets, (1) the edges, (2) the nodes and (3) the paths. > Consider the following three graphs: > x x x > / \ / \ > x x x---x x---x Are you joking? We are talking about paths representing real numbers. There is no real number which simutaneously has a 1 and a 0 at the same place. Your example is totally useless. > > It is quite similar with your tree. The union of all finite trees gives > a tree with all the nodes and edges, but not all the paths of the complete > tree. Only if you insist that there are real numbers which simultaneously have different numerals at the same position. > While the union of the sets of edges and nodes behave normal, the > union of the sets of paths is *not* the set of paths of the complete tree. No? Who told you so? Or is this your own fantasy? There are infinite paths because the paths are not finite. You should know from set theory: An infinite sequence is a sequence which is not finite. > And you are arguing about the union of the sets of paths. Of course, I am arguing about the union of the set of all path which really exist in a binary tree. This union is an infinite set of paths (although the set of paths in each finite tree is finite), and there is no hint on uncountability. Take the most right path: It is the union of all numbers 0.111...1 with n "1". This union is the number 0.111.... Similarly the union of all initial segments of a well-ordered countable set is this countable set. The union is not an element of the initial segments. But even in infinity, there is only "one" union and not uncountably many unions. > > > > > > You are wrong. sqrt(2) has a pretty good representation: "sqrt(2)". > > > > > > > > That is a name. Name ist Schall und Rauch. > > > > > > In the same way '2' is a name, '10' is a name. '10' is clearly a name > > > for a quantity that depends on the context. In the same way 'sqrt(2)' > > > is the name for a quantity that depends on the context. For instance, > > > the wedge that is used for the quantity 7 in Hyderabad Arabic is used > > > for the quantity 8 in Devanagari and for the quantity 6 in Javanese. > > > > Therefore these wedges are not numbers but only names which can mean > > anything we define. > > Right. And so is every notation of numbers. Not the notations below. > Numbers are not concrete > entities, they are abstract entities. No. Numbers are very concrete. Abstraction can be introduced but need not, at least not for small numbers. > And we name them by symbols or > strings of symbols according to particular convention. > > > But in Devanagari and Hyderabad Arabic and Javanese it is clear what we > > mean by > > > > |||||| > > ||||||| > > |||||||| > > Well, even I do not know. I would say either 8 or 7. Do you need new spectacles? > But try that in > a culture that has no idea about abstract entities. Yes, there is one > such culture, a tribe of Indians along the Amazone. They can not count, > and do not count, and do not understand counting at all. They will understand, at least by experiment, oo ooo _____ ooooo > In their > culture only concrete entities are acceptable (and that also only if > the person telling about it has personal experience with it). That is an advantage over people who see different identical trees. Regards, WM
From: mueckenh on 6 Jan 2007 13:14 Dik T. Winter schrieb: > In article <1168002324.590103.255240(a)s34g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1167859855.107241.239690(a)k21g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > That is precisely the infinite string of finite digit indexes of > > > > irrational numbers and similarly the infinite string of finite nodes of > > > > infinite paths. But one has no infinite paths in the tree? > > > > > > Not in the union of finite trees, but it is in the union of the full tree. > > > The two are not the same. > > > > Name a level or a node or and edge which are in one of the trees only. > > Why should I do that? In order to prove that there is evidence for at least one path being not in both trees. > I can name a *path* that is in the complete infinite > tree but not in the union of finite trees. Without different nodes that is an empty assertion. > > > It is equal. What Virgil is stating is that > > > N in U{n in N} {1, 2, 3, ..., n} > > > but that N is not one of the subsets used in the union. > > > > But, according to your point of view, N exists as the union of all > > finite sequences {1, 2, 3, ..., n}. Why do infinite paths not exist as > > the union of all finite paths? > > How many times do I have to explain that to you? Use the definition of > the union of an infinite collection of sets. If an element is in one > of the sets from the infinite collection it is in the union, if it is > in none of the individual sets it is also not in the union. Do you claim that the union of all finite trees is not an infinite tree? Why do you deny the same for the paths? The infinite paths in the union of trees are the unions of the respective finite paths. So much should be clear to every reader. What not has been clear, until most recently, is that these unions cannot be more abundant than the finite paths. The unions can at most be as abundant as the finite paths. That is why you try to come up with your fairy tales of different path-systems in identical trees. > > You were talking about the union of sets of finite paths. Asking for > an infinite path in that union is the same as asking for an infinite > number in the (particular) union of sets of finite numbers. I do not ask for an infinite path in that union. I claim that the union constitutes infinite paths. (That is obvious.) And I claim that these infinite paths cannot be more than the elements of the union. (That is obvious too.) Regards, WM
From: imaginatorium on 6 Jan 2007 13:19
Andy Smith wrote: > >> If you have an idealised bouncing ball that loses 1/2 > >> its height on each bounce and takes 1/2 the time, > >> how many bounces has it done in 2.00.. seconds? > >> Hasn't it achieved an actual infinity of bounces as a > >> finished thing? > > >It bounces infinitely many times. I don't know why you insist on adding > >the meaningless phrase "as a finished thing". But if you number the > >bounces sequentially, then each bounce gets a finite number. > > I am sure that you are right. To my uneducated mind such a thought > experiment demonstrated the ball metaphorically > counting up to omega,... What do you mean by "metaphorically" counting up to omega? Does it mean: "You can't count up to omega, but if you could, this would be what happens?" Or do you think you actually _could_ count up to omega? Here's the problem, and this is also why your "...0101" isn't a natural number: Suppose you start counting at 1, followed by 2, and so on in a way you're doubtless familiar with, soon reaching 57, 143, and way beyond. Do you notice that the decimal representations of the numbers you are counting all have a left end and a right end. 57 has 5 on the left end and 7 on the right end, as does 56334593302334955896669549050340349569667055434954886660970656549456747 (which you are not going to reach in practice, but that doesn't affect the discussion, does it?) [uh, surplus of question marks here] Do you see that the process of counting one of these two-ended numbers after another never ends? There is never a two-ended number such that you can't count on beyond it. Thus by any normal understanding of "counting" you will never reach anything like ...0101. "Omega" (in some loose sense) is certainly not a "number" with a two-ended decimal representation - actually it isn't in any normal sense anything like a "number" with a one-ended decimal representation, but even if you thought it was, you haven't demonstrated how to get to it by counting. > I will go and read a book and not waste your time further. That's perhaps a good idea - but you could always try reading the responses to the last 26 times someone explained in sci.math what is wrong with the idea you posted. I'm afraid it isn't new. Brian Chandler http://imaginatorium.org |