Cantor Confusion
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From: David Marcus on 5 Feb 2007 16:42 imaginatorium(a)despammed.com wrote: > I'm enjoying Dik's mini-review - I hope he will tell us the error in > chap. 7, and the two in chap. 8; perhaps at the end. Yes, I hope so, too. -- David Marcus
From: David Marcus on 5 Feb 2007 16:46 Andy Smith wrote: > I was just commenting that the adjectives "potentially" and "actually" > have their history (I think) in an Aristotelian philosophical > perspective of infinity, which is what Cantor tipped over. I think the > lay view (mine until recently) is still Aristotelian - infinity > exemplified by a never ending type of process. My knowledge of the history is certainly limited, but I believe that mathematicians in the 1800's and before were discussing the differences between potential and actual infinity. I doubt they were just doing this because they had read Aristotle. -- David Marcus
From: Lester Zick on 5 Feb 2007 17:30 On Mon, 5 Feb 2007 16:46:43 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Andy Smith wrote: >> I was just commenting that the adjectives "potentially" and "actually" >> have their history (I think) in an Aristotelian philosophical >> perspective of infinity, which is what Cantor tipped over. I think the >> lay view (mine until recently) is still Aristotelian - infinity >> exemplified by a never ending type of process. > >My knowledge of the history is certainly limited, but I believe that >mathematicians in the 1800's and before were discussing the differences >between potential and actual infinity. I doubt they were just doing this >because they had read Aristotle. You might be surprized. In fact without claiming to be a history expert either I wouldn't be a bit surprized if this was exactly what they were doing. ~v~~
From: mueckenh on 6 Feb 2007 04:41 On 4 Feb., 19:25, Virgil <vir...(a)comcast.net> wrote: > > > > The pathlength is not a natural number but it is a number (by > > > > definition), namely omega. You see that there is no infinite set N > > > > without an infinte number in it. > > > > As omega is not member of N I don't see that. > > > The total pathlength is the union of all single pathlengths. > > If all those separate pathlengths are finite ordinal numbers and there > are more than any finite number of them, then the union is the first > limit ordinal. So if there are infinitely many pathes of lengths 1, then the union of all of them is the first limit ordinal? You should distinguish between "how many" and "how large". In the EIT 1 11 111 .... there is no question of how many, because we could also consider a system with infinitely many lines like 1 11 11 11 111 111 111 111 .... but never getting more than 3 symbols in one line. To obtain the "number" omega, we need infinitely large lines. The EIT shows: There is no element of the diagonal unless there is a corresponding line. If there are actually infinitely many elements in the diagonal, then there must be actually infinitely many lines (which you would agree to) and actually infinitely many columns (which you deny for reasons beyond any logic, because this is the proof of nonexistence of actual infinitely many *finite* lines). > > The total > > pathlength cannot be omega without omega being a pathlengt (in the > > union). > > False! It is quite possible for a union not to be any one of the sets > being unioned. Interesting. There is some additional ghost? > > But omega cannot be in the union of finite pathlengths. Therefore there cannot be an actually infinite pathlength. > > It can be, and is, everywhere except possibly in WM's weird world. The pathlengths are corresponding to the lines in the EIT. If there is omega as a pathlength then there is omega as a line in the EIT. > > > "Infinite pathlength" means > > only that every finite pathlength is surpassed by another finite > > pathlength. > > Which requires a path with no end. No. In particular because there cannot be such a path. But if you assume its existence, then you see that the reals are countable or that identical nodes yield different path systems, an idea which presumably only you can utter. Regards, WM
From: mueckenh on 6 Feb 2007 04:42
On 4 Feb., 19:49, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > > On 4 Feb., 14:34, Franziska Neugebauer <Franziska- > > Neugeba...(a)neugeb.dnsalias.net> wrote: > >> mueck...(a)rz.fh-augsburg.de wrote: > >> > On 2 Feb., 12:10, Franziska Neugebauer <Franziska- > >> > Neugeba...(a)neugeb.dnsalias.net> wrote: > >> >> mueck...(a)rz.fh-augsburg.de wrote: > >> >> > On 2 Feb., 02:42, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > >> >> [...] > >> >> >> However, when we construct the path: > >> >> >> p = {n_1} U {n_1, n_2} U {n_1, n_2, n_3}, ... > >> >> >> we get as path: > >> >> >> {n_1, n_2, n_3, n_4, ...} > >> >> >> the path length in this case is *not* a natural number. It is > >> >> >> the cardinality of N. > > >> >> > The pathlength is not a natural number but it is a number (by > >> >> > definition), namely omega. You see that there is no infinite set > >> >> > N without an infinte number in it. > > >> >> As omega is not member of N I don't see that. > > >> > The total pathlength is the union of all single pathlengths. > > >> The path-length of the infinite path is the supremum of the set of > >> paths-lengths of all finite paths. The latter is the supremum of the > >> set of natural numbers. This supremum exists. > > > No. What does it consist of? > > It "consists" of (i.e. contains as members) every and all natural > numbers, each of which is finite. > > > A supremum exists in many places but not here. > > Not where? > > > The simplest reason is that omega - n = omega for all n in N. > > Where did you get that from? Reference? EB? You could even figure it out by yourself. Regards, WM |