Cantor Confusion
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From: mueckenh on 11 Jan 2007 16:44 Virgil schrieb: > 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. It is always true. If N is not actually infinite, then there are no actually infinite strings, i.e., in particular there are no irrational numbers. > > > > 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. I corrected that already: 0 is not a natural number. > > The union of binary trees is defined as the union of levels. > > Not in general. But here. > > 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. But here. > > > 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 union is the set of infinite paths. > > 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 The union is the infinite ordinal. Regards, WM
From: mueckenh on 11 Jan 2007 16:46 Virgil schrieb: > 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. An infinite union of finite sets is an infinite object. Regards, WM
From: mueckenh on 11 Jan 2007 16:52 William Hughes schrieb: > > > 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. No. It is an attribute 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. Then use the name "largest line". > > > > > > > > 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. So it is. > > In either case "the line" is not L_D. In both cases there is a largest line, namely the largest line which can be shown to exist or the largest line which has been shown to exist. The first one is obviously a contradiction, it refers to actual infinity. Hence, our only choice is the second.> > > > 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". Not individual lines, but largest lines. Regards, WM
From: mueckenh on 11 Jan 2007 17:01 Franziska Neugebauer schrieb: > > There are mathematicians greater than you and any living set > > terrorists which are convinced of the opposite. > > I don't keep company with terrorists. Pardon, a typing error. I meant set-theorists, of course. Excuse me. > > >> 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. Yes, but I am not a God. > > >> >> > 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. Even that is impossible without time. > > >> >> 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. They were teachers too. >I suppose the deeply buried truths of ancient > mathematicians will remain inaccessible to me. If there are any at all. It is so easy: Potential infinite sets are finite. But their maximum is not fixed. Regards, WM
From: Virgil on 11 Jan 2007 17:38
In article <1168550591.250119.271450(a)p59g2000hsd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > [...] > > > Conclusion: > > > Every finite binary tree contains a finite set of path. > > > > True. > > > > > The countable union of finite sets is countable. > > > > True. The union of countably many countable sets is countable. > > Yes, that is with AC. But even without AC the countable union of finite > sets is countable > > > > > The set of paths is countable. > > > > 1. The set of all rooted _finite_ paths in an infinite tree is > > countable. The union of all finite paths is exactly this set of all > > rooted finite paths. > > Yes, of infinitely many paths, each of which has infinitely many nodes. WM says "Yes, but in the sense of No!" Which ones of those specifically "rooted finite paths" does WM claim contain infinitely many nodes? > Why do you call the set {{1}, {1,2}, {1,2,3}, ...} which is equivalent > and even isomorphic to the set{1, 2, 3, ...} = N finite (finite path)? Each of the sets {1}, {1,2}, {1,2,3}, etc., is both a member and a subset of N, but N is not. > > The set of all natural numbers can be an infinite number omega, but the > set of all initial segments of N cannot be an infinite segment? > > Even the diagonal of the EIT is no longer infinite, if it is > inconvenient for set theory? As properly described, that diagonal is actually infinite however inconvenient that may be for WM's version of set theory. > > > > > 2. To "unary represent" every real in [0, 1] > > You cannot "unary represent" every real in [0, 1]. You can unary > represent every natural number, and you can unary represent real > numbers which contain only numerals a_n = const for every n in N > (behind the decimal point), like 0.111 or 0.1111111. In addition you > can unary represent omega or N = {1, 2, 3, ...}, namely as 0.111... = > {0.1, 0.11, 0.111, ...}. Notice that this set does not contain infinite > strings of 1. Then why does WM keep claiming that it does for trees? That is precisely what he claims for trees, that the concatenation of infinitely many finite members into a set somehow morphs into an infinite member of that set. |