Cantor Confusion
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From: Virgil on 6 Dec 2006 15:37 In article <1165403952.300542.55170(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Bob Kolker schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > There is no reason to give up continuity for applied mathematics. Ony > > > those who want to learn the real truth may bother. > > > > And what is the "real truth"? > > For instance the fact that the cardinal number of the set of all even > natural numbers cannot be larger than every even natural number, and, > as a consequence, the complete set of even natural numbers cannot > exist. That WM cannot conceive of something seems to justify WM's claim that it is inconceivable. Can WM conceive a new symphony? If not, it is inconceivable to him that anyone else can conceive one.
From: Virgil on 6 Dec 2006 15:38 In article <1165404884.005465.302970(a)16g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: > > > If you think sets grow, then you do not understand set theory. > > I know that in modern set theory sets do not grow. But I heavily doubt > the relevance of modern set theory. We heavily doubt the relevance of WM's opinions about modern set theory, or about much of anything else.
From: Virgil on 6 Dec 2006 15:43 In article <1165405610.352394.20240(a)16g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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? But WM's construction requires an infinite sequence of branches for each object to be paired with a path. And we already know that there are uncountably many such infinite sequences constructable from countably many branches(or nodes), so WM's pairing scheme is phony.
From: cbrown on 6 Dec 2006 17:30 mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > 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 > > 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. So now you return to claiming that your mapping T /is/ a surjection from edges onto paths. Perhaps there is some confusion regarding "what is the domain/range of T?". To clarify, let e be any edge; say the first branch to the left in your original diagram. According to your above statement, e is in the domain of T. Which path is T(e)? > > > > (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. Let p be the path representing the number 1/2. p is in the range of T because T is a surjection. Which 2 edges in the domain "the set of all edges" are mapped to p by T? (i.e., what is the pre-image T^-1(p)?) <snip> > > > 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. You haven't proved it until you provide a surjective function T : (edges -> paths). (And I am only interested in your "rational relation" argument). > > > > > 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. So you have a bijection S : (edges -> complete sets of shares of an edge). But given an edge e, what are the /elements/ of the set of shares, S(e)? How many elements does S(e) have? 1? 32? A countable number? An uncountable number? Cheers - Chas
From: Virgil on 6 Dec 2006 18:01
In article <4576BA70.3060402(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 9:20 PM, Virgil wrote: > > >> > Except that countable and uncountable coexist within the same set theory > >> > and rational and irrational coexist within the same real umber field. > >> > >> > >> Cantor's DA2 illustrates that there is no such field/list of real numbers. > > > > EB conflates "list" with "set". Nothing in any axiomatic set theory I am > > aware of requires sets to be lists, or even listable. > > Lists are countable sets. Not merely countable but "counted" in the sense of being the image of a specific function having the naturals as domain. > > > > >> Isn't this "coexistence" on the same low level of abstraction a basic > >> though hard to unveil intentional mistake by Dedekind? > > > > What "coexistence"? > > I refer to the necessary distinction between something concrete and the > abstract name of it. Real numbers are not really numbers in so far, they > do not have an accessable numerical address. It is only EB's mind that a 'number' has to have "an accessible numerical address". That alleged requirement is nowhere stated in mathematics, so is not a mathematical requirement. And nothing following from that false requirement is any part of mathematics either. > > > > >> Dedekind argued: As naturals can be extended to the integers in order to > >> allow subtraction and include negative numbers, and integers can be > >> extended to rationals in order to allow division and include fractions, > >> so rationals can perhaps be extended to reals in order to allow > >> non-linear operations and include irrationals. > > > > As Dedekind (and others) demonstrated precisely how to construct each of > > these extensions, his arguments conclude with Q.E.F. > > Did you mean Euclid's sentence quod errat demonstrandum (q.e.d.)? I mean Euclid's other sentence, as expressed in Latin: "Quod Erat Faciendum!" > > > What engineers cook up is not our problem. > > You are not a mathematician but as narrow-minded as is set theory. More of a mathematician than EB, which is all I need to be to refute EB's anti-mathematics. And being narrow minded, in the sense of being extremely literal minded, is a virtue in mathematicians which often irritates and upsets non-mathematicians in general, and anti-mathematicians like EB in particular, because they do not understand its purpose. |