Cantor Confusion
in [Math]
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From: mueckenh on 12 Apr 2007 16:27 On 12 Apr., 19:53, Virgil <vir...(a)comcast.net> wrote: > In article <1176397196.082457.230...(a)y5g2000hsa.googlegroups.com>, > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > On 11 Apr., 19:42, Virgil <vir...(a)comcast.net> wrote: > > > In article <1176298903.609533.227...(a)d57g2000hsg.googlegroups.com>, > > > > > > > Are the > > > > > > numbers represented by paths which are completely within the > > > > > > infinitely many levels of the tree or not? > > > > > > This sentence is quite difficult to pars. Yes, each real number is > > > > > represented by a path which is completely within the infinitely many > > > > > levels of the tree. > > > > > But not all real numbers coexist in the tree? > > > > Yes, all of them do at least once, and some twice. > > > But not side by side? More above or below each other? > > If WM visualizes paths as running from the root node downward, then sort > of side by side. When two paths represent the same binary rational (the > only reals having dual representation) there will be no other paths > "between" them. From the last node they have in common, one will branch > left once then forever branch right and the other will branch right once > then forever branch left, like 00111... and 0.1000... in binary both > represent the rational number 1/2. > > > > > > > Only a countable number > > > > of them is admitted simultaneously? > > > > All of them simultaneoulsy, and some twice. > > > But not side by side? > > Whyever not, at least for those represented twice? Do all paths exist side by side before the level number has increased to infinity or not? Regards, WM
From: mueckenh on 12 Apr 2007 16:30 On 12 Apr., 20:02, Virgil <vir...(a)comcast.net> wrote: > In article <1176397336.310678.13...(a)l77g2000hsb.googlegroups.com>, > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > On 11 Apr., 20:00, Virgil <vir...(a)comcast.net> wrote: > > > In article <1176300393.408149.165...(a)w1g2000hsg.googlegroups.com>, > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > On 11 Apr., 03:56, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > You think so. In that case you should throw away *all* of mathematics, > > > > > because there is (for instance) no direct proof that sqrt(n) is > > > > > rational > > > > > only if n is a square. When something is proven for an arbitrary > > > > > element > > > > > that means that is has been proven for all elements. > > > > > Yes, it has been proven for all elements (which exist). > > > > Name one which doesn't. > > > If I could name it, it would exist (at least as an idea). > > It follows that if you cannot name it , it does not exist, even as an > idea. > > > But we cannot name more than countably many. > > That is not because there cannot be more than countably many things but > because there cannot be more than countably many names. > > So that just as there can be more things than can be numbered by the > naturals, there can be more things than can be given all different > names. There can be more things than names, but not more names than names. Numbers are nothing unless being named. > > > > > > You cannot apply Cantor's proof to the whole diagonal. In particular > > > > > > because the diagonal does not exist. (It does not exist as a path > > > > > > separated from all other paths in the binary tree.) > > > > This fixation on "separation" is puerile. Every path in any binary tree > > > is separated from any other path from the children of some node onwards, > > > and that is as much separation as is needed to give each path a unique > > > identity. > > > But there are only countably many nodes! > > There are only countably many binary digit positions, but they are > enough to create uncountably many binary sequences. In the tree we see that every creation of another sequence requires a node. > > > > > > > > Again, your mantra. Proof, please. > > > > > If all paths exist simultaneously, then there must exist uncountably > > > > many in the infinite tree. > > > > Which is precisely the case! At least in mathematics. > > > But the tree is not in mathematics? > > If mathematically defined trees, finite or infinite, do not exist within > mathematics, why does mathematics bother to define them? And it does > bother to!- Why then do not all real numbers of [0, 1] exist side by side simultaneously in the so defined tree, while they exist in mathematics? Regards, WM
From: Virgil on 12 Apr 2007 17:38 In article <1176409628.060084.284540(a)p77g2000hsh.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 12 Apr., 19:53, Virgil <vir...(a)comcast.net> wrote: > > In article <1176397196.082457.230...(a)y5g2000hsa.googlegroups.com>, > > > > > > > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > > But not all real numbers coexist in the tree? > > > > > > Yes, all of them do at least once, and some twice. > > > > > But not side by side? More above or below each other? > > > > If WM visualizes paths as running from the root node downward, then sort > > of side by side. When two paths represent the same binary rational (the > > only reals having dual representation) there will be no other paths > > "between" them. From the last node they have in common, one will branch > > left once then forever branch right and the other will branch right once > > then forever branch left, like 00111... and 0.1000... in binary both > > represent the rational number 1/2. > > > > > > > > > > > Only a countable number > > > > > of them is admitted simultaneously? > > > > > > All of them simultaneoulsy, and some twice. > > > > > But not side by side? > > > > Whyever not, at least for those represented twice? > > Do all paths exist side by side before the level number has increased > to infinity or not? Only paths corresponding to binary rationals have a nearest neighbor, and then only on one side, like like 00111... and 0.1000..., corresponding to 1/2, will be neighbors, as no paths can come between them.
From: Virgil on 12 Apr 2007 17:48 In article <1176409858.797211.294160(a)p77g2000hsh.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 12 Apr., 20:02, Virgil <vir...(a)comcast.net> wrote: > > In article <1176397336.310678.13...(a)l77g2000hsb.googlegroups.com>, > There can be more things than names, but not more names than names. > Numbers are nothing unless being named. They are something if one wishes them to be. Whoever ruled that a number without a name does not exist? > > In the tree we see that every creation of another sequence requires a > node. Only in finite trees do leaf nodes biject with paths. In infinite trees there are no leaf nodes. > > > > > > > > > > > > Again, your mantra. Proof, please. > > > > > > > If all paths exist simultaneously, then there must exist uncountably > > > > > many in the infinite tree. > > > > > > Which is precisely the case! At least in mathematics. > > > > > But the tree is not in mathematics? > > > > If mathematically defined trees, finite or infinite, do not exist within > > mathematics, why does mathematics bother to define them? And it does > > bother to!- > > Why then do not all real numbers of [0, 1] exist side by side > simultaneously in the so defined tree, while they exist in > mathematics? And who says they do not? Every path defines a real in [0,1] as a convergent subsequence of 1/2 + 1/4 + 1/8 + ...+ 1/2^(n-1) + ... , in which the nth term is included if and only if the path branches left from its nth node.
From: mueckenh on 13 Apr 2007 05:07
On 12 Apr., 23:48, Virgil <vir...(a)comcast.net> wrote: > In article <1176409858.797211.294...(a)p77g2000hsh.googlegroups.com>, > > mueck...(a)rz.fh-augsburg.de wrote: > > On 12 Apr., 20:02, Virgil <vir...(a)comcast.net> wrote: > > > In article <1176397336.310678.13...(a)l77g2000hsb.googlegroups.com>, > > There can be more things than names, but not more names than names. > > Numbers are nothing unless being named. > > They are something if one wishes them to be. Whoever ruled that a number > without a name does not exist? In what form does it exist? In what form can it be used? > > > > > In the tree we see that every creation of another sequence requires a > > node. > > Only in finite trees do leaf nodes biject with paths. In infinite trees > there are no leaf nodes. But nodes of separation. > > > > > > > > > > > > > > Again, your mantra. Proof, please. > > > > > > > If all paths exist simultaneously, then there must exist uncountably > > > > > > many in the infinite tree. > > > > > > Which is precisely the case! At least in mathematics. > > > > > But the tree is not in mathematics? > > > > If mathematically defined trees, finite or infinite, do not exist within > > > mathematics, why does mathematics bother to define them? And it does > > > bother to!- > > > Why then do not all real numbers of [0, 1] exist side by side > > simultaneously in the so defined tree, while they exist in > > mathematics? > > And who says they do not? That one who says that there are less than all paths side by side in the tree. Regards, WM |