Cantor Confusion
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From: William Hughes on 11 Jan 2007 12:14 mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > No. But the property of being the greatest line can and does change. > > > > > > > > Irrelevant. The question is whether L_D exists. > > > > > > It does. Does the tallest man exist? When did it start, when did it > > > cease? > > > > Recall: > > > > L_D is a line that contains every element that can > > be shown to be in the diagonal. > > > > > > The "tallest man" is something that can change. L_D is a line. > > A line cannot change. The analogy is not valid. > > L_D is the name of a line, like a championship title. A championship title is not the name of a team. "The Chicago Bulls" is the name of a team. "The NBA champions" is not the name of a team. A championship title refers to a team, but the team it refers to can change. The name of a team refers to a team, but the team it refers to cannot change. L_D is the name of a line. The line it refers to cannot change. > > > > > It is possible to find L_D > > > > > > It is not assumed, but it is obvious that for every given set of > > > natural numbers there is one line containing it. > > > > But since the set of natural numbers can change > > this "one line" can change. It is not L_D. > > > > > > > > > > If you assume actual infinity then L_D > > > > does not exist. > > > > > > It does exist, in potential infinity. But it is not fixed. > > > > L_D is a line. A line is fixed. L_D does not > > exist. > > It is the line containing the whole set. Depends what you mean by "the whole set". If you mean every element of the diagonal that can be shown to exist, then "the line" does not exist. If you mean "all elements of the diagonal that have been shown to exist", then "the whole set" is something that can change and "the line" must be something that can change. In either case "the line" is not L_D. > > > > > > > > > Therefore you cannot assume actual infinity. > > > > > > > > You now admit that it is not possible to find L_D, > > > > > > In actual infinity (everything including L_D being fixed) it is not > > > possile to find L_D. > > > > And it is also not possible to find L_D in potential infinity. > > Let L_D go from 1 to oo. This statement is meaningless. L_D is a line. Lines do not "go from 1 to oo". - William Hughes
From: Franziska Neugebauer on 11 Jan 2007 12:23 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> >> > A potentially infinite quantity (set or not) is always >> >> >> > finite. >> >> >> >> >> >> There is no time in maths. >> >> > >> >> > So you write these letters and develop new ideas in zero time? >> >> >> >> My posts and my ideas do not take place *in* maths. They take >> >> place in the real _physical_ world. >> > >> > So does the contents of your posts and ideas, in part mathematics. >> >> This does not invalidate the fact that in mathematics there is no >> notion of time. > > A disadvantage or error The Mueckenheim way of admitting. > which need not perpetuate in eternity (et ultra). We need to perpetuate it so long and so far you do not refrain from claiming the nonesense. >>And hence in mathematics there are no temporal processes. This >> said "always" is not a intra-mathematically defined notion. > > There are mathematicians greater than you and any living set > terrorists which are convinced of the opposite. I don't keep company with terrorists. >> From this follows: "A potentially infinite quantity (set or not) is >> always finite" is a meaningless sentence. > > Everything you don't understand seems meaningless to you. You are unable to explain a mathematical meaning. I thought you were a teacher. >> >> > There is no existence outside of time. >> >> >> >> _Mathematical_ existence is not to be confused with physical >> >> existence. A mathematical entity x exists if there is a proof of >> >> "x exists". >> > >> > There is no proof existing outside space and time. >> >> This is an argument supporting which claim? > > Time is important in mahematics. Wishful thinking. >> >> As I have pointed out many times before: If and if so how the >> >> physical world determines our reasoning is off topic in sci.math. >> >> Please consult the neuro sciences groups for that complex of >> >> issues. >> >> >> >> This said your statement "A potentially infinite quantity (set or >> >> not) is always finite" makes no sense. >> > >> > Read the texts by Cantor and Hilbert given in my recent >> > contribution. >> >> I don't understand the fragments you have posted. > > Read the full texts. I gave the sources. You are the teacher. I suppose the deeply buried truths of ancient mathematicians will remain inaccessible to me. If there are any at all. >> They represent >> obviously merely a philophical discussion which do cannot understand. >> >> Perhaps you better reword the texts in your own lingo. If you want to >> state a different opinion to Cantor/Hilbert you should do so, too. >> >> Hence your sentence "A potentially infinite quantity (set or not) is >> always finite" still makes no sense. If it makes any sense you could >> explain that sense in your own words. > > Explain colour to a blind? "Wer Visionen hat, sollte zum Arzt gehen." F. N. -- xyz
From: Virgil on 11 Jan 2007 14:31 In article <1168511487.581261.39710(a)77g2000hsv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > Potential infinity is changing (growing) finity. > > > > Says who? The pope? > > Cantor and Hilbert (and everybody else who knows about it). Cantor and Hilbert are both dead and are no longer saying anything, and most of those who "know about it" nowadays say there ain't no such thing in any set theory. A set is either definitely finite with a definite finite size or definitely not with no finite size, at all, ever. Potential infiniteness is not a property that any set can have in any rational set theory. If one wishes to restrict one's attention to sets of finite (and fixed) sizes, one is perfectly free to do so, but in no case do sets vary in size as WM would have them do. WM is saying, in effect, that 2 + 2 = 5, for large enough values of 2.
From: Virgil on 11 Jan 2007 14:52 In article <1168511743.391940.58070(a)77g2000hsv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > >> Since there is no largest element in "potentially" infinite sets (in > > >> "actual/complete/finished", too) this sentence makes no sence at all. > > > > > > A potentially infinite quantity (set or not) is always finite. > > > > There is no time in maths. > > So you write these letters and develop new ideas in zero time? There is > no existence outside of time. > > > > > Therefore in a linearly ordered set here is a last element. Contrary > > > to the claim of set theorists, a set is not fixed in reality. > > > > When your arguments (by the way: What exactly are your arguments?) > > Here you can read it: > > Theorem. The set of real numbers in [0, 1] is countable. > > Lemma. > Each digit a_n of a real number r of the real interval [0, 1] in binary > representation has a finite index n. > r = SUM (a_n * 2^-n) with n in N and a_n in {0, 1}. This is only true if N is actually infinite, but is quite false otherwise. > > Proof. > A natural number n can be represented in a special unary notation: n = > 0.111...1 with n digits 1 (the leading 0. playing no role). Example: 1 > = 0.1, 2 = 0.11, 3 = 0.111, ... > In this notation the definition of the set of natural numbers, (1, 2, > 3, ...} = N, reads > > {0.1, 0.11, 0.111, ...} = 0.111.... (*) > > Note that also the union of all finite initial segments of N, {1, 2, 3, > ..., n}, is N = {1, 2, 3, ...}. Therefore (*) can also be interpreted > as union of initial seqments of the real number 0.111.... > > A real number r of the real interval [0, 1] can be represented as one > (ore two) path in the infinite binary tree. The set of all real numbers > r of the real interval [0, 1] is then given by the infinite binary > tree: > > 0. > / \ > 0 1 > / \ / \ > 0 1 0 1 > ................... > > A finite binary tree is the infinite binary tree, cut off below a level > n with n in N. > Here is a tree with two levels: > > 0. > / \ > 0 1 > / \ / \ > 0 1 0 1 > > namely level 1 and level 2. (The root at level 0 is conveniently not > counted, because 0 is not a real number.) The additive structure of the reals as a field requires that 0 be a real, so that, for example, 1 + (-1) will have a real value. Those who are so ignorant should curtail their attempts to correct what they do not even understand. > The union of binary trees is defined as the union of levels. Not in general. It is invalid unless one assumes that every tree shares the same root node and shares a lot of other stuff too. > The union of two or finitely many different finite binary trees simply > is the largest on. Not for disjoint trees. In that case the union is not a tree at all. > Taking the uninion of all finite binary trees, we get the complete > infinite binary tree with all levels. Since there exist disjoint binary trees, having no nodes, edges or paths in common, such unions are never trees. > All infinite paths representing > real numbers r of the real interval [0, 1] are in this union. We can > see this by the path always turning right, 0.111..., which is present > in the tree, according to (*). > > Conclusion: Every finite binary tree contains a finite set of path. The > countable union of finite sets is countable. The set of paths is > countable. The set of /finite/ paths in the union is countable. But when one takes the union of sets of finite paths one only gets finite paths in that union. There are no infinite paths in that union. The same thing happens with ordinals. When one takes the union of all finite ordinals (like unary trees), there is no infinite ordinal IN that union
From: Virgil on 11 Jan 2007 14:57
In article <1168511880.370120.180940(a)p59g2000hsd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Here is the formal proof: > > Theorem. The set of real numbers in [0, 1] is countable. Your proof was neither formal nor valid. Among other things you invalidly assume an infinite the union of finite sets must contain an infinite object, rather than merely containing infinitely many finite objects. |