Cantor Confusion
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From: Dik T. Winter on 10 Oct 2006 22:05 In article <MPG.1f93b0b45c6e7119896a5(a)news.rcn.com> David Marcus <DavidMarcus(a)alumdotmit.edu> writes: > Dik T. Winter wrote: .... > > Hm. I humbly submit that the probability for a particular rational number > > in the range [0,1) the probability to get it when doing a random choice > > is 0. > > Nevertheless, the sum of all the probabilities is 1. The sum of countably > > many 0's is not always 0. > > I don't follow. Usually, probability measures are countably additive. I just gave a counterexample. > As > for what Virgil wrote, presumably he meant for the sum in his definition > of B(t) to be the usual infinite sum from calculus/analysis (i.e., the > usual epsilon-delta definition). In which case, > > sum_{i=1}^infty 0 = 0. And see that there is a limit coming in. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 10 Oct 2006 22:18 In article <1160520935.025893.130590(a)h48g2000cwc.googlegroups.com>, "Albrecht" <albstorz(a)gmx.de> wrote: > I like to examine the idea of Russell Easterly - building a kind of > diagonal number on lists of natural numbers - in respect to this view. > First I review his idea: > > Let's have an arbitrary list of natural numbers: > > 1: a > 2: b > 3: c > ... > > with the numbers of digits of the numbers in the list: > > 1: m > 2: n > 3: o > ... > > Now we build the "diagonal number" d of the list as follows: > > We have a look on the first number of the list a which is build out of > m digits. We build a number with m+1 digits with the cipher 1. This > number is truely greater than a and therefore different from a. > It's the first approach to d. (E.g. a = 765 -> m = 3 -> d = 1111) > Now we have a look on the second entry of the list. The number b with n > digits. If n <= m we let d unchanged. If n > m we build the new number > d with n+1 digits, again with the cipher 1. > In this way we go through the list. > > This construction builds up a number d which is different from any > number of the list. Unless there is a natural number upper bound on the number of digits for numbers in the list, the m,n,o,..., the number of digits in the constructed what-ever-it-is will have to be greater than any finite number, and thus the constructed whatever-it-is will NOT be a number at all. And if there IS a natural number upper bound on the number of digits of listed numbers, it is easy to see that the list can only contain finitely many different numbers. > > Now we want to test this sentences: > > 1. Take ANY list of natural numbers. > 2. Show that a natural number can be defined/constructed from that > list. Russell's construction only works for essentially finite lists (finite when all duplications of values are omitted). > 3. Show that the natural number from step 2 is not on the list. > 4. Conclude that no list can contain all natural numbers. So we can conclude the o finite list can contain all natural numbers. Which is old news. > > 1.: Any lists are any finite and infinite lists of natural numbers > (Axiom of infinity). > 2.: The number d has a finite, integer difference to any number of the > list by construction. A number which could be written as the sum of two > natural numbers is a natural number too. > 3.: The number d is different to any number of the list by > construction. > 4.: The natural numbers are nondenumerable. > > > What's wrong with Russell's argument but right with Cantor's? See above. Either Russell's construction is not a number or the list contains only finitely many different numbers.
From: Virgil on 10 Oct 2006 22:24 In article <1160528819.116345.26700(a)m73g2000cwd.googlegroups.com>, cbrown(a)cbrownsystems.com wrote: > David Marcus wrote: > > David R Tribble wrote: > > > Dik T. Winter wrote: > > > >> And you also start with definitions, or a model. I did *not* state > > > >> that > > > >> it was difficult to define, or to make a model. But without such a > > > >> definition or model we are in limbo. I think other (consistent) > > > >> definitions > > > >> or models are possible, giving a different outcome. > > > > > > > > > > Virgil wrote: > > > >> Can you suggest one? One that does not ignore the numbering on the > > > >> balls > > > >> as some others have tried to do. > > > > > > > > > > David Marcus wrote: > > > > Models that ignore the number of the balls are certainly models. > > > > Whether > > > > they are reasonable translations of the original problem into > > > > Mathematics is a separate question. > > > > > > I tried modeling the problem with sets: > > > B_0 = { } > > > B_1 = B_0 U {1,2,3,...,10} \ {1} > > > ... > > > B_n = B_(n-1) U {10(n-1)+1,10(n-1)+2,...,10(n-1)+10} \ {n} > > > ... > > > > > > The resulting "balls in the vase at noon" would then seem to be B_w. > > > Which is problematic, since there is no ball labeled "w". So this is > > > probably a bad model of the problem. (I don't claim to be an expert.) > > > > I think you are still using the ball labeling. So, even if your model > > gave an answer for noon, I don't think it is what Virgil asked for. > > > > You could define B_w = {x | exists m such that for all n > m, x in B_n}. > > Another possibility is B_w = {x | for all m, there is an n > m such that > > x in B_n}. > > > > In both your and David Tribble's examples, I don't see any reference to > what time t step n occurs. This information is required to determine > "the number of balls at time t" for /any/ time t (not just noon). > > Cheers - Chas It only requires that each transition time, t_n, is before the next. t_{n+1}, and that all t_n's are before noon.
From: imaginatorium on 10 Oct 2006 23:57 Albrecht wrote: > Arturo Magidin wrote: > > In article <1159410937.013643.192240(a)h48g2000cwc.googlegroups.com>, > > <the_wign(a)yahoo.com> wrote: > > >Cantor's proof is one of the most popular topics on this NG. It > > >seems that people are confused or uncomfortable with it, so > > >I've tried to summarize it to the simplest terms: > > > > > >1. Assume there is a list containing all the reals. > > >2. Show that a real can be defined/constructed from that list. > > >3. Show why the real from step 2 is not on the list. > > >4. Conclude that the premise is wrong because of the contradiction. > > > > This is hardly the simplest terms. Much simpler is to do a ->direct<- > > proof instead of a proof by contradiction. > > > > 1. Take ANY list of real numbers. > > 2. Show that a real can be defined/constructed from that list. > > 3. Show that the real from step 2 is not on the list. > > 4. Conclude that no list can contain all reals. > > > > This summarization of the diagonal argument of Cantor seems to be > accepted by the most people in sci.math. > I like to examine the idea of Russell Easterly - building a kind of > diagonal number on lists of natural numbers - in respect to this view. > First I review his idea: > > Let's have an arbitrary list of natural numbers: > > 1: a > 2: b > 3: c > ... Aside: what do you understand a "natural number" to be? In mathematics, a natural number, when written out in (e.g.) decimal, is a string of digits with *two* ends. If you have different ideas on what your own personal "natural numbers" are, you may well be able to prove them uncountable. > > with the numbers of digits of the numbers in the list: > > 1: m > 2: n > 3: o > ... > > Now we build the "diagonal number" d of the list as follows: > > We have a look on the first number of the list a which is build out of > m digits. We build a number with m+1 digits with the cipher 1. This > number is truely greater than a and therefore different from a. > It's the first approach to d. (E.g. a = 765 -> m = 3 -> d = 1111) > Now we have a look on the second entry of the list. The number b with n > digits. If n <= m we let d unchanged. If n > m we build the new number > d with n+1 digits, again with the cipher 1. > In this way we go through the list. > > This construction builds up a number d which is different from any > number of the list. No it doesn't. It builds an unending string of digits. (Well, the English word "unending" means "having only one end, at the beginning, if that isn't too confusing. At any rate, there are not *two* ends.) An unending string of digits is not a representation of a natural number. > > Now we want to test this sentences: > > 1. Take ANY list of natural numbers. > 2. Show that a natural number can be defined/constructed from that > list. > 3. Show that the natural number from step 2 is not on the list. > 4. Conclude that no list can contain all natural numbers. > > 1.: Any lists are any finite and infinite lists of natural numbers > (Axiom of infinity). > 2.: The number d has a finite, integer difference to any number of the > list by construction. A number which could be written as the sum of two > natural numbers is a natural number too. > 3.: The number d is different to any number of the list by > construction. > 4.: The natural numbers are nondenumerable. > > > What's wrong with Russell's argument but right with Cantor's? In step 2 you have not created a natural number. The argument is just wrong. Brian Chandler http://imaginatorium.org
From: cbrown on 11 Oct 2006 02:02
Dik T. Winter wrote: > In article <virgil-372F10.17374709102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > > In article <J6w6LC.9rL(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: > > > In article <virgil-9E9CC6.02103209102006(a)comcast.dca.giganews.com> Virgil > > > <virgil(a)comcast.net> writes: > > > > > Dik T. Winter wrote: > > > ... > > > > > > The balls in vase problem suffers because the problem is not > > > > > > well-defined. Most people in the discussion assume some implicit > > > > > > definitions, well that does not work as other people assume other > > > > > > definitions. How do you *define* the number of balls at noon? > > > ... > > > > How about the following model: > > > > > > And you also start with definitions, or a model. I did *not* state that > > > it was difficult to define, or to make a model. But without such a > > > definition or model we are in limbo. I think other (consistent) definitions > > > or models are possible, giving a different outcome. > > > > Can you suggest one? One that does not ignore the numbering on the balls > > as some others have tried to do. > > That does not matter, nor is that the problem. You gave a model where you > find 0 as answer. I only state that I think there are also models where > that is not the answer. Why is a limit of the number of balls over time > not an answer? > > Let's give a simpler problem. At step 1 you add ball 1. At step n you > remove ball n-1 and add ball n (simultaneously, I presume). When you say "at step n", do you have some particular time t associated with that step? Cheers - Chas |