Cantor Confusion
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From: mueckenh on 14 Oct 2006 04:23 Virgil schrieb: > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > My purpose was to explain to you why your unreflected assumption is > > It is not > > contradictory to say that in a finite set of numbers there need not be > > a largest. It seems that this false assumption is one of the basic > > reasons for set theory. > > With any common meaning of "numbers" short of complexes, it is > contradictory in mathematics, whatever it may be in "Mueckenh"'s > philosophy. > While it may not be possible to determine which of that finite set of > numbers is largest, there has to be one. Why? Because a particle has to have a certain speed even if you cannot determine it? Or is there an axiom? Which one is it? Regards, WM
From: mueckenh on 14 Oct 2006 04:25 Virgil schrieb: > In article <1160648741.707624.62340(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1160577085.758246.228800(a)e3g2000cwe.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > If discontinuous functions were easily allowed everywhere, why then do > > > > you think that > > > > lim{n-->oo} n < 10 > > > > or > > > > lim{n-->oo} 1/n > 10 > > > > would be wrong? > > > > > > Since N is not normally considered to be a topological space, continuity > > > of functions with a non-topological domain N is a contradiction in > > > terms. > > > > Apply your knowledge to the balls of the vase. > > Which knowledge tells me that at noon each and every ball has been > removed from the vase. You are joking? > > > > > > On the other hand, limits of real sequences (functions from N to R) have > > > been quite adequately defined. One such definition is: > > > Give f:N --> R and L, then > > > lim_{n in N} f(n) = L (or lim_{n --> oo} f(n) = L > > > is defined to mean > > > For every real eps > 0, Card({n: Abs(f(n)-L) > eps}) is finite. > > > > For the vase problem with the number n(t) of balls in the vase after t > > transactions we can find always a positive eps such that for t > t_0: > > 1/n(t) < eps, hence n(t) larger than an arbitrary positive number. > > But that analysis does not carry beyond the times of transition, and > those times do not include noon or go past noon. > > You are assuming properties not given. If we can meaningfully calculate the limit of the harmonic sequence 1, 1/2, 1/3, ... --> 0 or the sum of the geometric series 1 + 1/2 + 1/4 + ... --> 2, then we can also calculate that at noon 1/n = 0. > > > > Therefore, your assumption of lim {t-->oo} n(t) = 0 is absurd. > > Your assumption that some ball that has been removed has not been > removed is even more absurd. Yes, that is true. Therefore the existence of all natural numbers can be excluded. To assume it is absurd. Regards, WM
From: mueckenh on 14 Oct 2006 04:30 imaginatorium(a)despammed.com schrieb: > David Marcus wrote: > > Dik T. Winter wrote: > > > In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > To inform the set theorist about the possible existence of sets with > > > > finite cardinality but without a largest number. > > > > > > Interesting but in contradiction with the definition of the concept of > > > "finite set". So you are talking about something else than "finite > > > sets". > > > > It would seem he is. I don't understand why people use words in non- > > standard ways without explaining what they mean. They are guaranteeing > > that no one will understand them. > > Several possible obvious answers. (BTW, it was fairly clear you were > new around here -- then you tried asking Ross Finlayson what he means. > Clinched it.) > > (a) The writer is playing a bizarre game of trollery. > (b) The writer is simply misinformed about the meaning of a particular > term. (Don't think this is common) > (c) The writer does not have the mental apparatus to understand a > formal argument, and therefore simply cannot comprehend the difference > between a number of statements of subtle difference. This seems to be > most common. For example, Mueckenheim - who astonishingly appears to > *teach* mathematics at some sort of college in Germany University of Applied Sciences, Augsburg. - 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. Regards, WM
From: mueckenh on 14 Oct 2006 04:33 Virgil schrieb: > > Consider the binary tree which has (no finite paths but only) infinite > > paths representing the real numbers between 0 and 1. The edges (like a, > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > edges is countable, because we can enumerate them > > > > 0. > > /a\ > > 0 1 > > /b\c /\ > > 0 1 0 1 > > ............. > > > > Now we set up a relation between paths and edges. Relate edge a to all > > paths which begin with 0.0. Relate edge b to all paths which begin with > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > a is inherited by all paths which begin with 0.00, the other half of > > edge a is inherited by all paths which begin with 0.01. > > One can relate one of the 'a' edges, say the left one, to all paths > beginning with 0.0 and the other 'a' edge, the right one with. the > string beginning 0.1 You misunderstood the notation (even after so many postings!). There is only one edge a taken as an example. The edge on the right hand side is not labelled. > > Then the left b edge is 0.00, the right b edge is 0.01, > the left c edge is 0.10, and the right c edge is 0.11 > > And at each level a left branching edge appends a 0 and a right > branching edge appends a 1 to the string > > Then every path truncated after a finite number of steps gives an edge > belonging to that path. > > >Continuing in > > this manner in infinity, we see that every single infinite path is > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, > > I do not see this at all, as "Mueckenh"'s method is not telling us > which edge a path takes at any node, so does not identify paths at all. > > A path consists of an infinite sequence of branchings, either left edge > or right edge at each node, and not merely fractional parts of edges. > > "Mueckenh" cannot show his "model" of a binary tree by any construction > strictly within ZFC or NBG. You have not understood it at all. So you casnnot judge. Regards, WM
From: William Hughes on 14 Oct 2006 04:33
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > ... > > > > > It is not > > > > > contradictory to say that in a finite set of numbers there need not be > > > > > a largest. > > > > > > > > It contradicts the definition of "finite set". But I know that you are > > > > not interested in definitions. > > > > > > We know that a set of numbers consisting altogether of 100 bits cannot > > > contain more than 100 numbers. Therefore the set is finite. The largest > > > number of such a set cannot be determined, as far as I know. > > > > There is a big difference between saying we do not know what > > the value of the largest element of a set is and saying that > > a set does not have a largest element. > > The set discussed above does not have a largest element. Every method > of representing its largest number can be surpassed by another one, I > believe. > Using 100 numbers here rather than 1 is simply obfuscation. You are saying the number N, does not have a size. Proof, if one gives a size to the number N, I can choose the number N again with a bigger size. Once you have chosen your set of 100 elements it will have a largest size. Questions as to the method of choice have no bearing on this. Placing you choice in the future makes the choice unknowable, it does not mean a choice will not be made. (If I choose a number N tomorrow it does not mean that the number N will not have a size, it means that I do not know, and cannot know, what the size of N will be. But it is quite possible to know both that N has a size and that the value of this size is unknowable). - William Hughes |