Cantor Confusion
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From: Andy Smith on 4 Jan 2007 18:56 I am (a physicist) suspicious of any arguments about infinite sets, which is why Cantor is useful to clarify ways of thinking about them. But what is wrong for this mapping of the real numbers in the interval [0,1] to the natural numbers: 1. All the reals can be represented as (countably) infinite binary sequences 2. All the numbers defined by the first N bits are uniquely indexed by the bit-reverse of their value 3. This is true for all N. If the argument for mapping the rationals onto the natural numbers is valid, then so is this? I am not giving up the day job yet.
From: imaginatorium on 5 Jan 2007 05:58 Andy Smith wrote: > I am (a physicist) suspicious of any arguments about infinite sets, which is why Cantor is useful to clarify ways of thinking about them. > > But what is wrong for this mapping of the real numbers in the interval [0,1] to the natural numbers: > > 1. All the reals can be represented as (countably) infinite binary sequences > > 2. All the numbers defined by the first N bits are uniquely indexed by the bit-reverse of their value > > 3. This is true for all N. > > If the argument for mapping the rationals onto the natural numbers is valid, then so is this? Don't quite follow 2. - something about "the first N bits"??? But anyway, what integer does your mapping map 1/3 (a real in [0,1]) to? Brian Chandler http://imaginatorium.org
From: mueckenh on 5 Jan 2007 07:22 Dik T. Winter schrieb: > > And my question is: Do these two trees, namely the complete tree and > > the union of all rational trees differ such that the one has edges or > > nodes which are missing in the other? > > No. > > But the distinction is apparently difficult (although about first year > at University for mathematics). The set of terminating binary expansions > is countable, the set of non-terminating binary expansions is no > countable. (You may replace terminating binary expansions with binary > expansions terminating with either a continuous stream of 0'z or of 1's.) But these streams are *not* present in all paths of the union of all rational trees! That has been overlooked, as it appears, even in the last years of university mathematics - in the last 130 years. 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? > > > > 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. But in Devanagari and Hyderabad Arabic and Javanese it is clear what we mean by |||||| ||||||| |||||||| Regards, WM
From: mueckenh on 5 Jan 2007 07:24 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > David R Tribble schrieb: > >> David R Tribble schrieb: > >> >> Well, now I'm confused. Could you provide an example of a natural > >> >> number that does not exist? > >> > > >> > >> mueckenh wrote: > >> > Take the first 10^100 digits of pi (if you can - but you cannot). > >> > It is impossible to bring this number to existence in the whole > >> > universe. > >> > >> Can you "bring into existence" the number 1? > > > > Here it is: > > > > . > > [X] a dot One clear representation of number 1, sufficient to identify it. Not only a name. Regards, WM
From: mueckenh on 5 Jan 2007 07:35
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. > > > > 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. You used to use ~iv: all the elements of the diagonal exist > 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. > > It is not necessary to assume the existence of the last digits > of all natural numbers (in unary representation). or to assume > the existence of all natural numbers. The fact that you can use A > along > with other assumptions to lead to another contradiction is > irrelevant, and does not make the contradiction between A No. But this contradiction arises from your assumption of actual existence (= completeness) of the diagonal. In potential infinity all these contradictions vanish. Regards, WM |