Cantor Confusion
in [Math]
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From: mueckenh on 21 Apr 2007 16:40 On 21 Apr., 20:10, Virgil <vir...(a)comcast.net> wrote: > In article <1177158861.675889.121...(a)y80g2000hsf.googlegroups.com>, > > > And what is the advantage does WM claim for "bunch" over "set". > Is not every "bunch" of paths a set of paths? > Yes, but "sets of sets" is a less clear notation than sets of paths or sets of path-bunches. > > > > There are bunches of paths like the bunch 0,1 which contains all paths > > from 0,1000... to 0.111... > > A searated path is a bunch of paths with only one element like > > 0,111... > > > In order to get two separateed bunches of paths, there is one bunch > > required which contains all these paths and one node to split the > > incomin bunch into wo bunches. > > But in a CIBT, all incoming and outgoing 'bunches", i.e., sets, of paths > are uncountable as sets (or as bunches, unless bunches are a lot > different than sets), Two bunches are twice as many as one bunch. There cannot be more bunches in the tree than nodes + 1. > > > > > The infinite tree contains all separated paths. Therefore there are as > > many nodes (minus one) required in the tree. > > Often claimed but never justified. > > There is no way that one node (or unit or whatever you wish to call it) > accomplishes the separation of one path from all others. It takes > infinitely many of nodes or whatevers. In any case the CIBT contains bunches which contain only single paths, if such paths exist anywhere. Regards, WM
From: William Hughes on 21 Apr 2007 16:55 On Apr 21, 4:33 pm, mueck...(a)rz.fh-augsburg.de wrote: > On 21 Apr., 15:00, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Apr 21, 8:30 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > On 19 Apr., 14:15, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > It is a set of nodes that separates a path, not a single > > > > node. There are an uncountable number of sets > > > > of nodes. So you do have enough nodes. > > > > First we should clear the notation: > > > There are bunches of paths like 0,1 which contains all paths from > > > 0,1000... to 0.111... > > > A separated path is a bunch of paths with only one element like > > > 0,111... > > > > In order to get two separateed bunches of paths, there is one bunch > > > required which contains all these paths and one node to split the > > > incomin bunch into wo bunches. > > > So a single node does not separate a single path. However, > > a set of nodes can. > > A single node increases the number of bunches of paths by 1. Even in Wolkenmuekenheim a bunch of paths is not a path. Each of these bunches contains an uncountable number of paths. All you show is that there are a countable number of bundles, each of which contains an uncountable number of paths. Each of these paths can be separated from all other paths by a set of nodes. There are uncountable many sets of nodes. There are enough nodes to separate uncountable many paths. - William Hughes
From: Virgil on 21 Apr 2007 17:37 In article <1177187620.795895.100950(a)b58g2000hsg.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 21 Apr., 15:00, William Hughes <wpihug...(a)hotmail.com> wrote: > > On Apr 21, 8:30 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > On 19 Apr., 14:15, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > It is a set of nodes that separates a path, not a single > > > > node. There are an uncountable number of sets > > > > of nodes. So you do have enough nodes. > > > > > First we should clear the notation: > > > There are bunches of paths like 0,1 which contains all paths from > > > 0,1000... to 0.111... > > > A separated path is a bunch of paths with only one element like > > > 0,111... > > > > > In order to get two separateed bunches of paths, there is one bunch > > > required which contains all these paths and one node to split the > > > incomin bunch into wo bunches. > > > > So a single node does not separate a single path. However, > > a set of nodes can. > > A single node increases the number of bunches of paths by 1. > > > > There are uncountable many sets of nodes. > > That is completely irrelevant. Not when so many of those sets of nodes each increase that number of "bunches", i.e., sets o paths, by 1. > > > > Therefore: > > > > There are enough nodes to separate uncountable many paths. > > No. A single node increases the number of bunches of paths by 1. And each such bunch is a set of uncountably many paths. > > > > We have to settle the question of whether there are enough nodes > > to separate uncountable many paths, before we go on to > > discuss the question of whether they in fact do separate uncountable > > many paths. > > This question has been settled. Then why is WM claiming one thing while everyone else is claiming a different thing about how it "has been settled"? WM is trying to beg the question. It is not "settled" unless either (1) everyone agrees, which is not the case, or (2) the majority rules, in which case WM loses. So that either WM concedes that it is not settled or he concedes that it has been settled against him by the majority. > Every single node increases the number > of bunches of paths by 1. The number of bunches surpassing the number > of nodes is given by Infinite trees have infinitely many non-leaf levels, and every subset of the set of all such levels determines the unique path that turns left at those levels and right at all others. So the number of paths equals the number of subsets of the set of all non-leaf levels. The set of non-leaf levels in a CIBT is countably infinite so that its power set, and the equivalent set of paths, must be uncountable. Nothing WM can make up about "bunches" can invalidate that argument.
From: Virgil on 21 Apr 2007 17:45 In article <1177187749.234029.162730(a)y5g2000hsa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 21 Apr., 19:57, Virgil <vir...(a)comcast.net> wrote: > > In article <1177158654.185563.309...(a)o5g2000hsb.googlegroups.com>, > > > > > > > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 19 Apr., 14:15, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > It is a set of nodes that separates a path, not a single > > > > node. There are an uncountable number of sets > > > > of nodes. So you do have enough nodes. > > > > > First we should clear the notation: > > > There are bunches of paths like 0,1 which contains all paths from > > > 0,1000... to 0.111... > > > A separated path is a bunch of paths with only one element like > > > 0,111... > > > > > In order to get two separateed bunches of paths, there is one bunch > > > required which contains all these paths and one node to split the > > > incomin bunch into wo bunches. > > > > > The infinite tree contains all separated paths. Therefore there are as > > > many nodes (minus one) required in the tree. > > > > Non sequitur. > > > > There is no proof that in an infinite binary tree, at least as defined > > in mathematical terms, the cardinality of the set of nodes is as large > > as the cardinality of the set of paths. > > Every node increases the number of path-bunches by 1. There are at least infinitely many paths, and every subset of the set of paths is a "bunch" of paths, so there are automatically uncountably many bunches. In a CIBT, (1)the set of non-leaf levels, L, is infinite, and (2) the set of paths is equinumerous with the power set of L Therefore the cardinality of the set of paths is equal to that of P(L) and greater than that of L
From: Virgil on 21 Apr 2007 17:59
In article <1177188039.701180.173860(a)y5g2000hsa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 21 Apr., 20:10, Virgil <vir...(a)comcast.net> wrote: > > In article <1177158861.675889.121...(a)y80g2000hsf.googlegroups.com>, > > > > > > And what is the advantage does WM claim for "bunch" over "set". > > Is not every "bunch" of paths a set of paths? > > > Yes, but "sets of sets" is a less clear notation than sets of paths or > sets of path-bunches. Not to me. Those who understand sets are not so easily confused by them as WM seems to be. > > > > > > > There are bunches of paths like the bunch 0,1 which contains all paths > > > from 0,1000... to 0.111... > > > A searated path is a bunch of paths with only one element like > > > 0,111... > > > > > In order to get two separateed bunches of paths, there is one bunch > > > required which contains all these paths and one node to split the > > > incomin bunch into wo bunches. > > > > But in a CIBT, all incoming and outgoing 'bunches", i.e., sets, of paths > > are uncountable as sets (or as bunches, unless bunches are a lot > > different than sets), > > Two bunches are twice as many as one bunch. Then does WM argue that the bunch of odd naturals together with the bunch of even naturals is twice as much as the bunch of all naturals? For FINITE bunches all of the same cardinality one can say that "Two bunches are twice as many as one bunch." But not necessarily otherwise. > There cannot be more > bunches in the tree than nodes + 1. Perhaps not in the wilds of WMville, but we are no longer stuck in the sticks. > > > The infinite tree contains all separated paths. Therefore there are as > > > many nodes (minus one) required in the tree. > > > > Often claimed but never justified. > > > > There is no way that one node (or unit or whatever you wish to call it) > > accomplishes the separation of one path from all others. It takes > > infinitely many of nodes or whatevers. > > In any case the CIBT contains bunches which contain only single paths, > if such paths exist anywhere. Such singleton sets exist, but are only "separated" from their complimentary sets of paths by infinite sets of nodes (any infinite subset of the node set of the one path being separated). |