Cantor Confusion
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From: imaginatorium on 18 Oct 2006 23:49 mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > imaginatorium(a)despammed.com schrieb: > > > > > > > > But the function of balls/numbers removed from the vase is a > > > > > continuously (stepwise) increasing one, containing all natural numbers > > > > > at noon? > > > > > > > > Uh, yes, unless I mysteriously misunderstand you... If takenout() is a > > > > function from time to the power set of the integers (i.e. it maps to a > > > > set of integers) then each natural number m is included in the set that > > > > takenout() maps to from time = -1/m. So by time zero, all natural > > > > numbers are included. > > > > > > > > Was there a question with that? > > > > > > And this result would change, if the numbers of the balls were > > > exchanged, for instance multiplied by 10 after having been inserted? > > > > Can you clarify what "exchanged" means? Obviously, if you do exactly > > the same thing with exactly the same balls, it makes no difference what > > is written on them, > > That is an astonishing statement, but I welcome it and agree. I expect all the non-cranks would agree with my statement, and find it astonishing. Cranks like yourself occasionally agree with all sorts of things, more or less by accident - so what? > Why do you speak of exactly the same balls, however? Because that's what I meant. > They are all indistinguishable like atoms. There are only the numbers. I think it's a standard assumption that when a thought-experiment refers to a macro-world object, it is precisely _not_ "like an atom". If the balls themselves are thus indistinguishable, it is a different thought-experiment, in which the operations are just "put a ball in" "take a ball out", and the number remaining at noon is completely undefined. In that case, if the numbers are capable of being labelled, then yes, it makes a difference. > > > but perhaps that is not what you mean. > > Take off balls 10, 20, 30, ... instead of 1, 2, 3, ..., or change the > numbers by multiplying them with 10. If you take out different (numbered) balls, obviously this makes a difference. If you just rewrite the number on the ball as something different, obviously it makes no difference. (Who knows, the English label for the number 10 might be graphically identical to the Martian way of writing one hundred.) Brian Chandler http://imaginatorium.org
From: David Marcus on 19 Oct 2006 00:09 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > However, you wish to do more. You want to show > > > > > > that claiming "N does not have an upper bound and > > > > > > N exists as a complete set" leads to a contradiction.] > > > > > > > > > > That is true too. And it is easy to see: If we define Lim [n-->oo] > > > > > {1,2,3,...,n} = N, then we can see it easily: > > > > > > > > > > For all n e N we have {2,4,6,...,2n} contains larger natural numbers > > > > > than |{2,4,6,...,2n}| = n. > > > > > There is no larger natural number than aleph_0 = |{2,4,6,...}|. > > > > > Contradiction, because there are only natural numbers in {2,4,6,...}. > > > > > > > > You appear to have written the following: > > > > > > > > Let N be the set of natural numbers. For all n in N, > > > > > > > > 2n > |{2,4,6,...,2n}| = n, > > > > > > No. I have written the following: For all n e N we have {2,4,6,...,2n} > > > contains larger natural numbers than |{2,4,6,...,2n}| = n. I did not > > > explicitly mention 2n. > > > > > > > > n < |{2,4,6,...}| = alpheh_0, > > > > > > > > {2,4,6,...,2n} is a subset of N. > > > > > > > > I follow this. But, you have the word "contradiction" in your last > > > > sentence. Are you saying there is a contradiction in standard > > > > Mathematics? If so, what is it? I don't see it. > > > > > > By induction we find the larger the set the larger the number of > > > numbers contained in the set and surpassing its cardinality. The > > > assumption that the infinite set would not contain such numbers > > > neglects the question of what kind of numbers can increase the > > > cardinality without increasing the number sizes. > > > The problem is the same, in principle, as the vase and its balls. > > > > You didn't mention 2n, but you said "larger natural numbers than n". 2n > > is one of those numbers that is in the set and is larger than n, so I > > wrote "2n". Is that not what you meant? > > It is not what I meant, because the main observation is that the number > of numbers larger than n grows with n. It is just the vase-ball > situation. In addition there is no largest natural number as could be > suggested by the last number 2n of the finite sequences. > > > > As for your final paragraph, I'm sorry, but I can't follow it. Please > > start with the following and add whatever is missing to give the full > > proof. > > > > Let N be the set of natural numbers. For all n in N, > > > > |{2,4,6,...,2n}| = n, > > there exists m in {2,4,6,...,2n} such that m > n, > > and the number of these numbers m grows with n. There is no finite > natural number n, such that the number of m's in the set > {2,4,6,...,2n} is larger than the number of m's in the set > {2,4,6,...,2n, 2(n+1)}. > or briefly: The cardinality of the set {n,n+1,n+2,...,2n} is strictly > increasing. > > Therefore, in the limit {2,4,6,...} there are infinitely many finite > natural numbers m > |{2,4,6,...}| Once again you use words and phrases that are not part of standard mathematics, i.e., "in the limit {2,4,6,...} there are infinitely many finite natural numbers m > |{2,4,6,...}|". What does "in the limit {2,4,6,...}" mean? You said (repeatedly) that standard mathematics contains a contradiction, so please state the contradiction in standard mathematics. If you use new terms, please define them. -- David Marcus
From: David Marcus on 19 Oct 2006 00:12 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > Virgil schrieb: > > > > > > > > > > > My sympathies to his poor students. > > > > > > > > > > I will tell them your ideas about the vase and then ask them about > > > > > their opinion. But don't forget: They are not yet spoiled by what you > > > > > call logic. > > > > > > > > > > > > > > - plainly cannot > > > > > > > > comprehend the difference that swapping quantifiers makes. He cannot > > > > > > > > comprehend that there might be a difference between the significance of > > > > > > > > "every" in "Every girl in the village has a lover" and "John makes love > > > > > > > > to every girl in the village". > > > > > > > > > > > > > > Is the Imaginator too simple minded to understand, or is it just an > > > > > > > insult? The quantifier interchange is impossible in general, but it is > > > > > > > possile for special *linear* sets in case of *finite* elements. > > > > > > > > > > > > For example? > > > > > > > > > > > > Does "Mueckenh" claim that, say, > > > > > > "For every natural n there is a natural m such that m > n" > > > > > > and > > > > > > "There is a natural m, such that for every natural n, m > n" > > > > > > are logically equivalent? > > > > > > > > > > > > All the elements are finite and linearly ordered. > > > > > > > > > > The second statement is obviously wrong, because there cannot be a > > > > > natural larger than any natural. > > > > > The quantifier exchange however is possible for sets of finite numbers > > > > > n the following form: > > > > > "For every natural n there is a natural m such that m >= n" > > > > > and > > > > > "There is a natural m, such that for every natural n, m >= n" > > > > > This natural m is not fixed. It is the largest member of the set > > > > > actually considered. > > > > > > > > Please let people know when you are not using standard terminology and > > > > when you do this, please define your terms. What does it mean to say a > > > > natural number "is not fixed"? > > > > > > One cannot know it, cannot call it by its name, but it is provably > > > present. You should be familiar with this from of existence. It is like > > > the well-order of the reals: present but very. > > > > I'm sorry, but "cannot call it by name" is not a "form of existence" > > that I've seen in any math book I've ever read. Please define what you > > mean using standard mathematics. > > Didn't you read in your books that the well order of the reals, for > example, cannot be defined better than the size of this number? > The state of existence is: Existence proven but construction or definition > impossible. The word "constructible" has a standard meaning. The phrase "cannot call it by name" does not. If you mean a number cannot be constructed, then please say so. Don't make up new words for standard concepts. -- David Marcus
From: David Marcus on 19 Oct 2006 00:17 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > > > First you refer to a "relation between paths and edges". > > > > > > Correct. > > > > > > > Then you say > > > > the "edge is related to a set of path". > > > > > > Is there any contradiction? > > > > Using standard mathematical terminology, there most certainly is a > > contradiction. You have apparently given two different descriptions as > > to what the elements of your "relation" are. In one, the elements are > > paths, but in the other they are sets of paths. A path is not a set of > > paths. Please state the definition of your "relation" clearly. If you > > use a word in other than its standard mathematical meaning, then please > > give a mathematical definition. > > Read this again: > > > > I warned you that this point is new: The edges are split in shares of > > > 1/2, 1/4, and so on. But when fractions were introduced in mathematics, > > > most people may have had the same problems as you today. I am sure you > > > can understand it from the written text above. (Many others have > > > already understood it.) > > If a fraction of an edge is related to a path, which is really new, as > far as I know, then the whole edge is related to a set of paths. > Otherwise it would be meaningless to use fractions. This new technique > was exactly described in my original text. Unfortunately, it was described in a way that I can't understand it. A wild guess on my part is that you mean to set up a correspondence between edges and sets of paths. However, it shouldn't be necessary for me to guess what you mean. Please state your entire argument again using consistent, well-defined terminology. -- David Marcus
From: David Marcus on 19 Oct 2006 00:20
mueckenh(a)rz.fh-augsburg.de wrote: > That is correct. That is precisely the reason why the natural numbers > appear to be infinitely many. There are at most 10^100 different > numbers but there are far larger values. The set is finite, but has not > a largest element. In standard mathematics, a finite set of natural numbers has a largest element. So, what are you actually saying? Are you saying that you don't like standard mathematics? Or, are you stating a result in standard mathematics, but using non-standard terminology? -- David Marcus |