Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: David Marcus on 16 Oct 2006 18:01 Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > > stephen(a)nomail.com wrote: > > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > > stephen(a)nomail.com wrote: > > > > > > >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > >> > > > >>>Dik T. Winter wrote: > > > >> > > > >>>>In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > > > >>>>Han.deBruijn(a)DTO.TUDelft.NL writes: > > > >>>> > Virgil schreef: > > > >>>>... > > > >>>> > > I do not object to the constraints of the mathematics of physics when > > > >>>> > > doing physics, but why should I be so constrained when not doing physics? > > > >>>> > > > > >>>> > Because (empirical) physics is an absolute guarantee for consistency? > > > >>>> > > > >>>>Can you prove that? > > > >> > > > >>>Is it possible to live in a (physical) world that is inconsistent? > > > >> > > > >> Perhaps. How could we know? > > > > > > > How can we know, heh? Can things in the real world be true AND false > > > > (: definition of inconsistency) at the same time? > > > > > > > Han de Bruijn > > > > > > What does it mean for a thing in the real world to be true? > > > How do you know if a thing in the real world is true? > > > > > > Consider the twin slit experiment. Is the fact that none of > > > the following accurately describe the situation an inconsistency? > > > a) the photon goes through one slit > > > b) the photon goes through both slits > > > c) the photon goes through neither slit > > > > In Bohmian Mechanics (and similar theories), the photon goes through > > only one slit. Physicists could learn something about logical thinking > > from mathematicians. > > Sure, theories. Can't you talk about something else but "theories"? In the real world, the photon undoubtedly goes through one slit. However, discussing physics with you is undoubtedly even more useless than discussing mathematics. -- David Marcus
From: David Marcus on 16 Oct 2006 18:20 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? 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, n < |{2,4,6,...}| = alpheh_0, {2,4,6,...,2n} is a subset of N. -- David Marcus
From: David Marcus on 16 Oct 2006 18:24 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. -- David Marcus
From: David Marcus on 16 Oct 2006 19:30 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > I gave my translation of the ball and vase problem. I don't see any > > contradictions that follow from it. If you say it implies two > > contradictory conclusions, please state them and your proof. > > What was your result? How many balls are in the vase at noon according > to your translation? I gave my translation of the problem into Mathematics. I didn't give an answer to the mathematics problem. You say there are two contradictory answers. Please state the two answers and the proof of each. Here again is my statement of the problem in Mathematics: For n = 1,2,..., define A_n = 12 - 1 / 2^(floor((n-1)/10)), R_n = 12 - 1 / 2^(n-1). For n = 1,2,..., define a function B_n by B_n(t) = 1 if A_n < t < R_n, 0 if t < A_n or t > R_n, undefined if t = A_n or t = R_n. Let V(t) = sum{n=1}^infty B_n(t). What is V(12)? -- David Marcus
From: William Hughes on 16 Oct 2006 19:30
mueckenh(a)rz.fh-augsburg.de wrote: > Alan Morgan schrieb: > > > > >> As I have inductively gone through the entire list of balls introduced > > >> into the vase and found that each of them has been removed before noon, > > >> why should stating that trivial fact be considered a joke? > > > > > >But you cannot go inductively through the cardinal numbers of the sets > > >of balls in the vase? They are 9, 18, 27, ..., and, above all, we can > > >show inductively, that this function can never decrease. > > > > You think that's bad? I have an even simpler situation! Add one ball > > at 1 minute to noon, another ball at half a minute to noon, another > > at 1/4 minute to noon, and so on. The number of balls in the vase before > > noon is always finite, but somehow, miraculously, at noon the number of > > balls in the vase becomes infinite. When, oh when, does that transition > > from finite to infinite happen? > > > > I submit that this is just as wierd a result as the original problem. > > Weird is that adding 9 balls instead of 1 per transaction leads to zero > balls. > Weird is that taking off 1 ball per transaction leads to all balls > taken off and no ball remaining, if the enumeration is 1,2,3,... but to > infinitely many balls remaining, if the enumeration is 10, 20, 30, .. . > This in particular is weird because there is a simple bijection between > 1,2,3,... and 10, 20, 30, ... > Right. No matter which balls you pick you are going to remove an infinite number of balls. So the number of balls you remove does not matter. Which balls you remove does matter. - William Hughes |