Cantor Confusion
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From: Tonico on 11 Oct 2006 04:29 Albrecht a écrit : >> > > 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 ******************************************************************** I wonder whether some posters here are just that stupid, or else they are so devoid of any mathematical education and yet they have the nerve to post in this group anyway, or else they just are completely empty of any intellectuaal honestity...*sigh*...as Woody Allen already said once: "His lack of education is more than compensated for by his keenly developed moral bankruptcy." Anyway, and since Albrecht FINALLY posted a link to a mathematician trying his best to knock Set Theory foundations (he's basically ranting on many things, among them his stubborn refusal to admit that infinite sets are VERY FORMALLY and properly defined in set theory and in many books; perhaps the author had a VERY bad mathematical education in his birth country Canada. But he rises some interesting points, and perhaps I shall come back to that paper of his in another thread)...anyway, since Albrecht finally posted that link which I asked for, I break my promise not to participate in this thread and answer briefly his last nonsense about the ridiculous Easterly R.'s construction: (1) First, unlike Cantor's construction, with Easterlye's construction you can ACTUALLY reach a point where the number you're constructing is ALREADY in the list. For example, if the list begins with 5, 11, ... already in the second step your constructed number is in the list; (2) Second: if the given list is actually infinite, the constructed "number"...is NOT a number! We already had a very annoying, at first but later it was kindda joke, crank in a spanish maths group that insisted in trying to construct "natural numbers" with an infinte number of digits. This is exactly what happens with Easterly's Method (damn! It even sounds serious...) if the given list is actually infinite, e.g. if it contains all the natural numbers; (3) Why then the method of Cantor works? Very simple: because real numbers between 0 and 1, say CAN have an infinite number of decimal digits to the right of the point. What to do, things are like this! Now. either Albrecht is unable to understand the difference between infinite decimal development and an infinite string of digits, the last of which could be called &#$$&#*, but NOT a natural number, and then all is hopeless, or else he does understand the difference but then he lacks the intellectual integrity to confess this or,. at least, to remain silent. Peano's Axioms (some get a seizure when they see "axiom" on writing, as if they were asked to believe and humbly accept the DOGMA "gods exist" or else they'll be BBQed by some mathematical inquisition. Don't worry kids: we have not such a thing) are, so far, a consistent set of axioms (or else PROVE otherwise, don't merely rant about it), and if you don't like them you may try to establish new rules (we call them "axioms" in mathematics) and see if something new, interesting and sound comes up. Either way, this already is too weary. I propose to begin a new thread. Regards Tonio
From: Tonico on 11 Oct 2006 04:35 Albrecht a écrit : .......................... > 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 ******************************************** Unbelievable...Albrecht just wrote "The diagonal number is only finite in difference to any natural number by construction. How could this numbe be not natural?" LOL! Are you serious: So you ALREADY establish "the diagonal number", begging in a rather lame way its existence, and then you ask your detractors "how this number can't be a natural number"?!?! "This is a car. If it weren't a car, then this car wouldn't be a car. So how this car can't be a car?" Common, IT is a car...or not?" <<== like this is the kind of nonsense you're posting, and I bet you don't even get ashamed...**sigh**... Tonio
From: imaginatorium on 11 Oct 2006 05:04 Albrecht wrote: > 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. You "say", but what you say is not true. If we consider just the reals in the interval [0, 1), then every one has a decimal representation as a string of digits with just one end (at the left). The diagonal process constructs exactly that again - a string of digits with just one end (at the left); therefore it is indeed one of the reals. As can be seen by anyone with the intelligence required to understand elementary mathematics, there is indeed a contradiction. > 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? No. At each stage in the process of constructing this diagonal sequence this may be true, but it is not true of the entire unending sequence of digits. Otherwise, I'll give you a natural number, 57 say, and you tell me the finite difference between 57 and your unending string of digits. I guess you cannot understand that the sequence of two-ended natural numbers that starts 1, 2, 3, 4, ... 57, ... goes on without having an end. (If so, I'm afraid the problem is at your end. 'ouch!') Brian Chandler http://imaginatorium.org
From: mueckenh on 11 Oct 2006 05:36 David Marcus schrieb: > > You wrote that "A covers B" means that A has at least as many bars as B. > > Does "S is completely covered by at least one element of the infinite > set of finite unary numbers" mean that S is covered by an A that has a > finite number of bars? If S = 0.111... had only finite positions then it was covered by finite numbers. All of them are in the list (these are unary representations of the natural numbers) 1 = 0.1 2 = 0.11 3 = 0.111 .... If a set of finite numbers of the list cover some number x of the list, then always already one list number is sufficient to cover number x. If some number y (in unary representation) is not in the list, then it cannot be covered by any list number. Then its positions are not well defined. This is the case with S. Regards, WM
From: mueckenh on 11 Oct 2006 05:56
David Marcus schrieb: > > I am sure you are able to translate brief notions like "to enter, to > > escape" etc. by yourself into terms of increasing or decreasing values > > of variables of sets, if this seems necessary to you. Here, without > > being in possession of suitable symbols, it would become a bit tedious. > > Yes, I can translate it myself. However, that would only tell me how I > interpret the problem. Hasn't it become clear by the discussion? I use two variables for sequences of sets. Further I use a function. I use the natural numbers t to denote the index number. The balls are simply the natural numbers. I speak of "balls" in order to not intermingle these numbers with the index-numbers. The set of balls having entered the vase may be denoted by X(t) with. So we have the mathematical definition: X(1) = {1,2,3,...10}, X(2) = {11,12,13,...,20}, ... with UX = N There is a bijection between t and X(t). The set of balls having left the vase is described by Y(t). So we have the mathematical definition: Y(1) = 1, Y(2) = {1,2}, ... with UY = N There is a bijection between t and Y(t). And the cardinal number of the set of balls remaining in the vase is Z(t). So we have the mathematical definition: Z(t) = 9t with Z(t) > 0 for every t > 0. There is a bijection between t and Z(t). > Until you tell me how you would translate it, I > don't know how you are interpreting the problem. Before we can draw any > mathematical conclusions, we need to know what mathematical problem we > are discussing. If you prefer, I could offer a translation and you could > tell me if it is what you mean. Fine. Regards, WM |