Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 24 Jan 2007 10:56 On 24 Jan., 00:42, "David R Tribble" <d...(a)tribble.com> wrote: > mueckenh wrote (2007-01-18): > > > 1. The union of all finite trees is an infinite tree. > > 2. Every finite tree contains only a finite set of paths. > > 3. The countable union of all paths of the finite trees is therefore the > > countable union of all finite paths. > > 4. The countable union of all finite paths is in the union of all finite trees. > > 5. The "complete" tree containing all paths is identical to the union of > > al finite trees, with respect to nodes and edges. > > 6. Identical trees cannot contain different sets of paths. > > 7. Therefore, both trees contain the same set of paths. > > 8. Therefore the "complete" set of all path is countable. > > 9. Therefore the set of all real numbers is countable. > > 10. Therefore ZFC is inconsistent.Just wondering if you're still using this argument. > > If so, perhaps you can explain how you manage the leap > from (8) to (9). It looks like you're talking about the union > of all finite-length trees (whatever that means) {a,b,c} U {a,b,c,d,e,f,g,h,} = {a,b,c,d,e,f,g,h,} Ordering of the nodes is defined by the type of tree. > in (1) through > (8), then at (9) it looks like you somehow conclude that the > set of all finite trees is equivalent to the set of all reals. Paths in a tree are always to be understood as maximum paths. No path ends before the tree has ended. Paths in the tree containing all nodes are all we can use to represent a real number. 1) Every complete infinite binary tree T (containing all nodes and edges) contains all paths. 2) The union tree T(oo) of all finite trees is well defined (as I have shown elsewhere) and yields the complete infinite binary tree containing all nodes and edges: T = T(oo). 3) The union of all finite trees includes the union of all nodes and, with it, the union of all such subsets which are paths (because every path is a well defined subset of the set of nodes if the structure of the tree is well defined). 4) The set of paths in T(oo) is a subset of the countable set of finite sets of all paths in the finite trees. 5) A countable union of countable sets is a countable set (according to ZF with AC). ==> The set of all path is countable. (==> The real numbers are countable.) Going on, we can say: 6) T(oo) = T contains only finite paths. 7) T(oo) = T contains all paths including all infinite paths. ==> There are no infinite paths. (There are no irrational numbers.) Nothing further remains to say. Regards, WM
From: mueckenh on 24 Jan 2007 11:02 On 24 Jan., 01:07, Virgil <vir...(a)comcast.net> wrote: > Of course, "winning" does not require that one convinces the crank > himself of anything, it only requires convincing the vast majority of > lurkers of the crank's crankhood. If the vast majority consists of lurkers believe in the finished infinity, this result is excluded. > > In which task the crank often unconsciously cooperates in particular if he sees different sets of paths realized by same sets of nodes and edges. But I cannot believe, in fact, that you argue unconsciously. Regards, WM
From: stephen on 24 Jan 2007 11:39 Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > In message <1169646430.885558.7060(a)m58g2000cwm.googlegroups.com>, > imaginatorium(a)despammed.com writes >> >>Andy Smith wrote: >>> In message <ep7kn4$79j$2(a)mailhub227.itcs.purdue.edu>, Dave Seaman >>> <dseaman(a)no.such.host> writes >>> >On Wed, 24 Jan 2007 09:48:01 GMT, Andy Smith wrote: >>> >> In message <9b2er2p5taadlea28n6fb3g4nuvmqeijhs(a)4ax.com>, G. Frege >>> >><nomail(a)invalid.?.invalid> writes >>> >>>On Tue, 23 Jan 2007 23:34:03 GMT, Andy Smith >>> >>><Andy(a)phoenixsystems.co.uk> wrote: >> >><snip> >> >>> If you have a transcendental, you need to specify an infinite number of >>> bits to distinguish it from the set of all alternative transcendentals. >>> You specify reals as a Cauchy sequence, which unambiguously points >>> towards the point, but the point itself needs an infinite number of bit >>> positions. But you can't label an infinite number of bit positions - you >>> need to have all of the bits as a completed set to define the real - and >>> that is not a finite number. >> >>What do you mean by a "completed set"? Is an unending sequence such as >>the following a "completed sequence"? >> >>1, 11, 111, 1111, 11111, 111111, 1111111, ... >> >>If "completed" means "having an end", then this is _not_ "completed", >>because it continues without end. But if "completed" means "complete" - >>that is, that there is nothing "missing" - then the sequence above >>includes every two-ended string of 1s, even though there are an >>unending number of them. Do you disagree? > Um, well I thought there was a distinction between the sequence > 1,2,3,4,n,.. and {1,2,3,4,n,..}. All the terms in the sequence are > finite, but the set of all of them is infinite. The key distinction between a sequence and a set is that a sequence is ordered. In both the sequence 1,2,3,4, ... and the set {1,2,3,4, ... } each element is finite, and both the sequence and the set are infinite. > And that was what I was > trying to say, was that a real point is not the sequence > b0,b0b1,b0b1b2,.. but determined by > {b0,b1,b2,...} 9and that actually was the whole point about indexing > the reals, which frankly I wish I hadn't mentioned by now). A real number is determined by a sequence. Sets have no order, and do not contain duplicates. The set {0,1,2,3,4,5,6,7,8,9 } is the same as the set { 9,8,7,6,5,4,3,2,1,0}. The set {0,1,1,0,1} is the same as the set { 1,0,0,0,0,1} is the same as the set {0,1}. There is no way to represent real numbers using a simple set containing only digits. Sequences can be represented as sets. The sequence 0,1,1,0,1 can be represented as the set { (0,0), (1,1), (2,1), (3,0), (4,1) }. Stephen
From: Andy Smith on 24 Jan 2007 11:59 stephen(a)nomail.com writes > >The key distinction between a sequence and a set is that a sequence >is ordered. In both the sequence 1,2,3,4, ... and the set {1,2,3,4, ... } >each element is finite, and both the sequence and the set are >infinite. > >> And that was what I was >> trying to say, was that a real point is not the sequence >> b0,b0b1,b0b1b2,.. but determined by >> {b0,b1,b2,...} 9and that actually was the whole point about indexing >> the reals, which frankly I wish I hadn't mentioned by now). > >A real number is determined by a sequence. Sets have no order, >and do not contain duplicates. The set {0,1,2,3,4,5,6,7,8,9 } >is the same as the set { 9,8,7,6,5,4,3,2,1,0}. The set {0,1,1,0,1} >is the same as the set { 1,0,0,0,0,1} is the same as the set {0,1}. >There is no way to represent real numbers using a simple set containing >only digits. > >Sequences can be represented as sets. The sequence 0,1,1,0,1 >can be represented as the set { (0,0), (1,1), (2,1), (3,0), (4,1) }. > Thanks for the clarification. I had previously mistakenly been using the expression "actually infinite" to mean the infinite sequence limit - a "completed infinity" as Aristotle wouldn't have approved of. But, anyway, you require an infinite number of bits to define the set of reals and so any indexing scheme also requires an infinite number of bits, and so any corresponding index is not a natural number. Crank, crank... -- Andy Smith
From: stephen on 24 Jan 2007 12:34
Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > stephen(a)nomail.com writes >> >>The key distinction between a sequence and a set is that a sequence >>is ordered. In both the sequence 1,2,3,4, ... and the set {1,2,3,4, ... } >>each element is finite, and both the sequence and the set are >>infinite. >> >>> And that was what I was >>> trying to say, was that a real point is not the sequence >>> b0,b0b1,b0b1b2,.. but determined by >>> {b0,b1,b2,...} 9and that actually was the whole point about indexing >>> the reals, which frankly I wish I hadn't mentioned by now). >> >>A real number is determined by a sequence. Sets have no order, >>and do not contain duplicates. The set {0,1,2,3,4,5,6,7,8,9 } >>is the same as the set { 9,8,7,6,5,4,3,2,1,0}. The set {0,1,1,0,1} >>is the same as the set { 1,0,0,0,0,1} is the same as the set {0,1}. >>There is no way to represent real numbers using a simple set containing >>only digits. >> >>Sequences can be represented as sets. The sequence 0,1,1,0,1 >>can be represented as the set { (0,0), (1,1), (2,1), (3,0), (4,1) }. >> > Thanks for the clarification. I had previously mistakenly been using the > expression "actually infinite" to mean the infinite sequence limit - a > "completed infinity" as Aristotle wouldn't have approved of. What do you mean by the infinite sequence limit? The limit of an infinite sequence diverges, does not exist, or it is finite. > But, anyway, you require an infinite number of bits to define the set of > reals and so any indexing scheme also requires an infinite number of > bits, and so any corresponding index is not a natural number. Crank, > crank... What do you mean by an indexing scheme? Each real number can be represented by an infinite number of bits. Each of those bits has a finite index. Stephen |