Cantor Confusion
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From: Franziska Neugebauer on 6 Dec 2006 06:37 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: >> mueckenh(a)rz.fh-augsburg.de writes: [...] >> I can reasonably think about the set of all natural >> numbers. Honest, I have no problem with it. > > You believe you could. That is a difference to "can". Other people > think they can reasonably think of having an immortal soul. By the way: Did you ever have problems in school? I think of literature, text interpretation or the like. You have to frequently "think" issues which are not "real" in the sense that "fictous" characters do "interact fictously". It is not uncommon that you have to discuss specific questions about those characters in order to obtain a satisfying school report regardless of whether you like it. F. N. -- xyz
From: mueckenh on 6 Dec 2006 06:46 cbrown(a)cbrownsystems.com schrieb: > > But a unit can be divided. You know what 1/2 is. And you know what 1/4 > > is. You know tat 1/2 + 1/2 = 1. > > > > I know that 1/2 of the length of a unit edge + 1/2 of the length of a > unit = 1 unit edge in length; but I don't know that 1/2 of an edge + > 1/2 of an edge = 1 edge; because I don't know what "1/2 of" an edge is. 1/2 edge is an entity which together with anther entity 1/2 edge can be summed up to yield 1 edge. > > Some things (unit lengths) can be divided in a sensible manner; and > some things cannot. Likewise, some things, such as /lengths/, can be > added so that 1/2 + 1/2 = 1; but 1/2 of the group of order 7 + 1/2 of > the group of order 7 is not the group of order 7; because "1/2 of" is > meaningless here. Half of a man is also meaningless (how should be divided? after division the man is dead!). Nevertheless we calculae with half men. Understood? > > Your argument seems to say something about lengths of paths and lengths > of edges; but not about number of paths and number of edges (where by > "number" I mean "cardinality"). If you have 27/2 half men, then you can be sure to have moe than 10 full men. > > > > > If you dislike the fractions only... > > > > > > I don't dislike fractions; some of my best friends are fractions. > > > > > > I simply don't see that you have produced, from the functions f, g, and > > > h, a surjective function T (edges -> paths). You appear to have no > > > response to the request that you produce one; and instead change the > > > problem. > > > > As those parts of an edge which are mapped on a path are not mapped on > > any other path, there is obviously a bijection, though not from > > undivided edges but from the shares of divided edges onto paths. > > > > Your original argument was: "Edges are countable. Paths are > uncountable. There exists a surjection T of edges onto paths; therefore > there countable >= uncountable; contradiction". > > Now you say that T is /not/ a surjection of edges onto paths, but some > other thing. I have two questions: > > (1) What is the range and domain of the bijection T you claim to have > provided? More exactly, when you say T maps "shares of divided edges" > one-to-one onto "paths", how do you characterize the set of "shares of > divided edges"? What exactly are the elements of this set? The domain is all edges. > > (2) How does the existence of T then lead to a contradiction? The number of full edges mapped on a path is larger than 1. > > (Prediction: you are somehow conflating the fact that lim n->oo sum > (over all edges e) g(p,e,n) = 2 with the (false) assertion that sum > (over all edges e) lim n->oo g(p,e,n) = 2.) No. > > > > So I take it you agree that your original argument is flawed? > > > > No. The necessity of as many edges as path is so obvious that this fact > > is impossible to overlook - once one has discovered it. > > > > If it is so obvious, you should, as a professor, also be able to prove > it; otherwise, it is simply your firmly held conviction. I have proved it by rational relatin and by a random mapping. > > > I add an appendix to one of my papers, where this is underlined I (here > > the arguing is based on nodes instead of edges, but that doesn't matter > 4> much): > > > > Your appendix fails to address the key question: what is the domain of > the function T? If e is an edge, what is the set of "shares of the > divided edge e"? What is your problem? The complete set of shares of one edge is the full edge. Regards, WM
From: Franziska Neugebauer on 6 Dec 2006 07:00 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: >> > Everybody knows what the number of ther EC states is. [...] > The number of EC states is "the number of EC states". This is hardly a definition. > It is simply a notion which can be equal to a natural number. Which may _evaluate_ to a number. Without explicitly or implicitly providing a context (year) there is no definite answer in terms of natural numbers. Mathematically the number of EC states can be modelled as _function_ of time. > The set of prime numbers does not contain the number 1. According to a widespread defintion of "prime number" the set of prime numers _refers_ _to_ a set which does not contain the number 1. > But once upon a time it did contain it. There may have been a time, when _a_ _different_ _set_ (one containing the 1) was _referred_ _to_ _by_ the named "set of prime numbers". This does not imply that the set of former times "has changed" in time. Only the naming has changed due to a changed definition. You may compare this with Gerhard Schr�der who did not undergo a gender transformation when Angela Merkel became chancellor in 2005. F. N. -- xyz
From: Franziska Neugebauer on 6 Dec 2006 07:30 mueckenh(a)rz.fh-augsburg.de wrote: > Bob Kolker schrieb: [...] >> Rational numbers are non-genuine. Nowhere in the physical world >> outside of our nervouse systems do they exist. > > The nervous system is a part of the physical world. Hence this issue is to be disussed in a physics or neuro sciences newsgroup. > The integers belong to the nervous system. The nervous system is generally not an issue in mathematics. You should discuss this in an appropriate neuro sciences newsgroup. > They would not exist without the nervous system. This is a miscomprehension. Mathematical numbers exist mathematically. If you want to discuss how abstract entities, i. e. naive numbers, "exist" in the sense of "are represented in the neural system" you should discuss this issue in an appropriate neuro sciences newsgroup. > And they cannot surpass its limits. That is the same case as > with the human soul. It is clear that the mind is heavily influenced > and personality can be changed by drugs and poison and by removing > parts of the brain. Juvpu qehtf naq cbvfbaf qb lbh gnxr? (abg nypbuby, jr xabj ...) Juvpu cnegf bs lbhe oenva unir orra erzbirq? (SCNR) > But some people nevertheless believe in the immortal soul. This is not a mathematical issue. You may discuss "immortality of the soul" in an appropriate (religous?) newsgroup. > I think that is just as strange as the belief in one's > own capability of imaging actual infinity. > And, remarkably, the creator of this theory believed in both, the > soul/God and actual infinity, It is remarkable that the former Generalfeldmarschall of Nazi Germany Hermann G�ring frequently spent his vacation in the Harz. Just like you (and possibly Cantor, too,) do. To me it is not clear whether you prefer to wander the trails of Cantor or that of G�ring. > trying to prove the latter by the existence of the former. Since mathematically there is not a "god" this "proof" was hardly a mathematical one. Let me summarize: You are mostly off topic. F. N. -- xyz
From: Franziska Neugebauer on 6 Dec 2006 07:36
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: >> > Everybody knows what the number of ther EC states is. [...] > The number of EC states is "the number of EC states". This is hardly a definition. > It is simply a notion which can be equal to a natural number. Which may _evaluate_ to a number. Without explicitly or implicitly providing a context (year) there is no definite answer in terms of natural numbers. Mathematically the number of EC states can be modelled as _function_ of time. > The set of prime numbers does not contain the number 1. According to a widespread defintion of "prime number" the set of prime numers _refers_ _to_ a set which does not contain the number 1. > But once upon a time it did contain it. There may have been a time, when _a_ _different_ _set_ (one containing the 1) was _referred_ _to_ _by_ the named "set of prime numbers". This does not imply that the set of former times "has changed" in time. Only the naming has changed due to a chance in what definition is accepted. You may compare this with Gerhard Schr�der who did not undergo a gender transformation when Angela Merkel became chancellor in 2005. F. N. -- xyz |