Cantor Confusion
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From: Franziska Neugebauer on 18 Jan 2007 08:04 mueckenh(a)rz.fh-augsburg.de wrote: > Carsten Schultz schrieb: >> > You assume that the union P_i of paths contains more paths than can >> > be constructed from finite initial segments? >> >> I do not assume anything. I just note that being a path in the union >> of the T_i and being an element of the union of the P_i are a priori >> different things and that you would have to prove their equivalence >> in your setting should you claim this equivalence. > > The union of all finite trees is an infinite tree. Definition of union of all finite trees and if necessary proof of existence? F. N. -- xyz
From: Andy Smith on 18 Jan 2007 10:15 mueckenh(a)rz.fh-augsburg.de writes > >Andy Smith schrieb: > >> I >> > > > > The union of all finite binary trees contains all levels >> > > > >which can be >> > > > > enumerated by natural numbers: >> > > > > >> > > > > 0 0. >> > > > > / \ >> > > > > 1 0 1 >> > > > > / \ / \ >> > > > > 2 0 1 0 1 >> > > > > ............................... >> > > > > >> >> Out of interest, aren't the set of all numbers defined by the union of >> all paths through a finite binary tree with N levels just all the >> numbers addressed by the first N bits? If so, why do you bother with >> the tree construction - does it have some special significance? > >The real numbers are represented as infinite paths in the "complete" >infinite tree. Some even twice. > >The union of all finite trees is an infinite tree. >Every finite tree contains only a finite set of paths. >The countable union of all paths of the finite trees is therefore the >countable union of all finite paths. >The countable union of all finite paths is in the union of all finite >trees. >The "complete" tree containing all paths is identical to the union of >al finite trees, with respect to nodes and edges. >Identical trees cannot contain different sets of paths. >Therefore, both trees contain the same set of paths. >Therefore the "complete" set of all path is countable. >Therefore the set of all real numbers is countable. >Therefore ZFC is inconsistent. I would have said that the set of all paths in a finite tree of depth N correspond 1:1 with the address range of N bits. An infinite tree corresponds to a number encoded in a countably infinite set of bits. Cantor's diagonalisation argument then applies. But, I think that there are other reasons for thinking that the reals are uncountable anyway. But I am not qualified to comment anyway. Regards -- Andy Smith
From: David Marcus on 18 Jan 2007 12:51 Andy Smith wrote: > mueckenh(a)rz.fh-augsburg.de writes > > > >Andy Smith schrieb: > > > >> I > >> > > > > The union of all finite binary trees contains all levels > >> > > > >which can be > >> > > > > enumerated by natural numbers: > >> > > > > > >> > > > > 0 0. > >> > > > > / \ > >> > > > > 1 0 1 > >> > > > > / \ / \ > >> > > > > 2 0 1 0 1 > >> > > > > ............................... > >> > > > > > >> > >> Out of interest, aren't the set of all numbers defined by the union of > >> all paths through a finite binary tree with N levels just all the > >> numbers addressed by the first N bits? If so, why do you bother with > >> the tree construction - does it have some special significance? > > > >The real numbers are represented as infinite paths in the "complete" > >infinite tree. Some even twice. > > > >The union of all finite trees is an infinite tree. > >Every finite tree contains only a finite set of paths. > >The countable union of all paths of the finite trees is therefore the > >countable union of all finite paths. > >The countable union of all finite paths is in the union of all finite > >trees. > >The "complete" tree containing all paths is identical to the union of > >al finite trees, with respect to nodes and edges. > >Identical trees cannot contain different sets of paths. > >Therefore, both trees contain the same set of paths. > >Therefore the "complete" set of all path is countable. > >Therefore the set of all real numbers is countable. > >Therefore ZFC is inconsistent. > > I would have said that the set of all paths in a finite tree of depth N > correspond 1:1 with the address range of N bits. > > An infinite tree corresponds to a number encoded in a countably infinite > set of bits. > > Cantor's diagonalisation argument then applies. But, I think that there > are other reasons for thinking that the reals are uncountable anyway. The diagonal argument is sufficient. But, there are other proofs. > But I am not qualified to comment anyway. You are more qualified than is WM. -- David Marcus
From: MoeBlee on 18 Jan 2007 13:08 Han de Bruijn wrote: > MoeBlee wrote: > > > [ ... ] And meanwhile, I always have my ears open for ideas not in > > the mainstream, including intuitionism, finitism, ultra-finitism, > > non-standard logics, paraconsistent logic, as well as the many > > philosophical approaches such as constructivism, realism, > > structuralism, fictionalism, and even as farflung as certain mystical > > views of logic and mathematics. (But that doesn't entail that I don't > > also exercise my prerogative to skewer postings by cranks.) > > How about materialism, engineering, applications, numerical analysis, > computer graphics? You have't seen anything of the latter kind, huh? There's nothing I wouldn't like to know more about. I do tend to prefer purely abstract mathematics to applied mathematics, though that is not a philosophical position so much as just a personal inclination. This subject is but a hobby for me, and I am not even talented in the subject, so, alas, my finite time and intellect prevent me from knowing about everything all at once, even as much as I would like to. > > I haven't found your questions to be stupid. But I do think your > > questions would benefit by being put in the light of mathematical > > definitions and some familiarity with certain basics of the subject. > > Especially the "light" is what bothers some of us .. A stunning rejoinder you have posted. MoeBlee
From: Virgil on 18 Jan 2007 15:23
In article <36389$45af2b2f$82a1e228$19589(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > MoeBlee wrote: > > > [ ... ] And meanwhile, I always have my ears open for ideas not in > > the mainstream, including intuitionism, finitism, ultra-finitism, > > non-standard logics, paraconsistent logic, as well as the many > > philosophical approaches such as constructivism, realism, > > structuralism, fictionalism, and even as farflung as certain mystical > > views of logic and mathematics. (But that doesn't entail that I don't > > also exercise my prerogative to skewer postings by cranks.) > > How about materialism, engineering, applications, numerical analysis, > computer graphics? You have't seen anything of the latter kind, huh? > > > I haven't found your questions to be stupid. But I do think your > > questions would benefit by being put in the light of mathematical > > definitions and some familiarity with certain basics of the subject. > > Especially the "light" is what bothers some of us .. > > Han de Bruijn We have noticed that you perfer darkeness. |