Cantor Confusion
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From: Virgil on 11 Oct 2006 02:37 In article <1160546562.540946.205860(a)e3g2000cwe.googlegroups.com>, cbrown(a)cbrownsystems.com wrote: > 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 Time at step n is t_n = 1/2^(n-1) minutes before noon.
From: Albrecht on 11 Oct 2006 03:20 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. Nobody posted first. Best regards Albrecht S. Storz
From: Albrecht on 11 Oct 2006 03:25 David Marcus schrieb: > 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 > > ... > > > > 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. > > > > 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. > > I don't follow. How do you know that the procedure that you gave > actually "defines/constructs" a natural number d? It seems that you keep > adding more and more digits to the number that you are constructing. What is the difference to the diagonal argument by Cantor? Best regards Albrecht S. Storz > > > 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? > > -- > David Marcus
From: Albrecht on 11 Oct 2006 03:32 imaginatorium(a)despammed.com schrieb: > 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. > I say about Cantors proof: In step 2 you have not created a real number. The argument is just wrong. I say about Russells proof: The diagonal number is only finite in difference to any natural number by construction. How could this number be not natural? Best regards Albrecht S. Storz
From: Albrecht on 11 Oct 2006 03:42
Virgil schrieb: > 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. But by construction the number must be finite since at every step it is build up by finite addition to a finite number. So we have to conclude that there is a finite number which is greater than any natural number. This result is as fantastic as Cantor's result about more than (countable) infinite many real numbers. Your intuition may confuse you. But it's not unusual in math that the results don't match the intuition, really. Best regards Albrecht S. Storz > > 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. |