An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: imaginatorium on 5 Nov 2006 14:04 Ross A. Finlayson wrote: > imaginatorium(a)despammed.com wrote: > ... > > > > The minimum number of faces is obviously 4; not less than 3 faces must > > meet each vertex, and there must be more than one vertex. I make a > > little list of the number of possibilities: > > > > 4: 1 (tetrahedron) > > 5: 2 (square pyramid, triangular prism) > > 6: (pentagonal prism, cube, etc. at this point, cheat > > http://www.research.att.com/~njas/sequences/table?a=944&fmt=4 ) > > 7: and so on > > > > Now I would notice that by "topologically distinct polyhedron" I am > > referring to a bounded geometrical object; not only bounded, but also > > discrete, in the sense that if I have one in my hand I know I can count > > the vertices. A bit of minor handwaving, and I would see that for any > > number of vertices there will be a limited number of possibilities for > > arranging that number of vertices into a polyhedron. So I know that I > > can pick an ordering scheme, and put all of the polyhedra in it. So I > > can "count" them, in the sense that I know that with my counting scheme > > there will not be a polyhedron that escapes it. I also notice that this > > counting sequence will never end, because there is no maximum to the > > number of vertices. After all, if there were, then given a polyhedron > > with that number of vertices I could simply pick any face, construct a > > pyramid on that face, and get a polyhedron with more than the supposed > > maximum number of vertices, which proves (by contradiction) that my > > sequence of polyhedra never ends. (This is all incredibly obvious, but > > I'm afraid I never know which incredibly obvious bits you still haven't > > grokked.) > > > > So with the constraints of the present audience, I would express this > > by saying that I can see I can "count" the polyhedra, a process which > > will account for every one of them, given enough time, but I can also > > see that the process of counting will never end. > > > > It's much messier, but I can do a similar thing for the polygons with > > vertices having integral x-y coordinates. > > > > How could I compare the two sets? I don't know - in both cases it's > > true that I can find a way of counting them, and that the counting will > > never end. Since it never ends, it's hard to see how I might think that > > the counting of the polygons was going to be over before the counting > > of the polyhedra, or vice versa. It's not even possible to say of a > > process that never ends that after so many counts the processs is "an > > appreciable way to completion", because there _is_ no completion. > > > > So my answer is rather limited: both sets are (ok, dammit, in normal > > words) countably infinite; there's no way obvious to me that I could > > regard either as "less" endlessly endless than the other. > > > > What's your answer? You appear to be the one claiming to have "numbers" > > for counting things when the counting never ends: do you have any here? > > Or are you happy to accept that these two sets, and the set of pofnats > > can all be put in 1-1 correspondence, or (in normal words again) have > > the same cardinality? Perhaps this particular fact raises no objections > > from your intuition module? > > > > Notice that of course there are other things one can say about these > > sets. For example, the number of polyhedrons with n edges [E(n)] is > > always less than the number with n vertices [V(n)] (= the number with n > > faces); I can see that E(n) < V(n-2), but "on average" these are > > close(?). Does the ratio V(n)/E(n) approach a particular value? I don't > > know. > > > > Brian Chandler > > http://imaginatorium.org > > Maybe n-gonometry would help, towards n-k-hedrometry. > > Re "countable uncountable", does it not seem that nested intervals must > apply to irrationals else it wouldn't? Ross, you seem a pleasant enough chap, but your posts always remind me of that wonderful scene in the Python film "Eric the Viking", in which Sekine Tsutomu, a Japanese radio personality I believe, gets a bit part as the slave master on the ship - thrashing his whip he pours out a torrent of abuse, totally unrelated to the words in the subtitles, and one of the rowers - says in best Python-cockney, "Well, it wouldn't be so baaaad, if we knew wot 'e was saying, would it?" OK. What is n-gonometry? What is "n-k-hedrometry"? What on earth does the oxymoron "countable uncountable" have to do with anything I said? (It's just a remnant of the original bit of nonsense from the ineducable German.) I can't parse the rest of the sentence, but I have no idea what it would mean for "nested intervals" to "apply" to irrationals. (Hang on, though, I vaguely recall that one of your claims clear enough to be wrong was some confusion about nested intervals. Who knows...) Brian Chandler http://imaginatorium.org
From: Ross A. Finlayson on 5 Nov 2006 16:14 imaginatorium(a)despammed.com wrote: > Ross A. Finlayson wrote: > > imaginatorium(a)despammed.com wrote: > > ... > > > > > > The minimum number of faces is obviously 4; not less than 3 faces must > > > meet each vertex, and there must be more than one vertex. I make a > > > little list of the number of possibilities: > > > > > > 4: 1 (tetrahedron) > > > 5: 2 (square pyramid, triangular prism) > > > 6: (pentagonal prism, cube, etc. at this point, cheat > > > http://www.research.att.com/~njas/sequences/table?a=944&fmt=4 ) > > > 7: and so on > > > > > > Now I would notice that by "topologically distinct polyhedron" I am > > > referring to a bounded geometrical object; not only bounded, but also > > > discrete, in the sense that if I have one in my hand I know I can count > > > the vertices. A bit of minor handwaving, and I would see that for any > > > number of vertices there will be a limited number of possibilities for > > > arranging that number of vertices into a polyhedron. So I know that I > > > can pick an ordering scheme, and put all of the polyhedra in it. So I > > > can "count" them, in the sense that I know that with my counting scheme > > > there will not be a polyhedron that escapes it. I also notice that this > > > counting sequence will never end, because there is no maximum to the > > > number of vertices. After all, if there were, then given a polyhedron > > > with that number of vertices I could simply pick any face, construct a > > > pyramid on that face, and get a polyhedron with more than the supposed > > > maximum number of vertices, which proves (by contradiction) that my > > > sequence of polyhedra never ends. (This is all incredibly obvious, but > > > I'm afraid I never know which incredibly obvious bits you still haven't > > > grokked.) > > > > > > So with the constraints of the present audience, I would express this > > > by saying that I can see I can "count" the polyhedra, a process which > > > will account for every one of them, given enough time, but I can also > > > see that the process of counting will never end. > > > > > > It's much messier, but I can do a similar thing for the polygons with > > > vertices having integral x-y coordinates. > > > > > > How could I compare the two sets? I don't know - in both cases it's > > > true that I can find a way of counting them, and that the counting will > > > never end. Since it never ends, it's hard to see how I might think that > > > the counting of the polygons was going to be over before the counting > > > of the polyhedra, or vice versa. It's not even possible to say of a > > > process that never ends that after so many counts the processs is "an > > > appreciable way to completion", because there _is_ no completion. > > > > > > So my answer is rather limited: both sets are (ok, dammit, in normal > > > words) countably infinite; there's no way obvious to me that I could > > > regard either as "less" endlessly endless than the other. > > > > > > What's your answer? You appear to be the one claiming to have "numbers" > > > for counting things when the counting never ends: do you have any here? > > > Or are you happy to accept that these two sets, and the set of pofnats > > > can all be put in 1-1 correspondence, or (in normal words again) have > > > the same cardinality? Perhaps this particular fact raises no objections > > > from your intuition module? > > > > > > Notice that of course there are other things one can say about these > > > sets. For example, the number of polyhedrons with n edges [E(n)] is > > > always less than the number with n vertices [V(n)] (= the number with n > > > faces); I can see that E(n) < V(n-2), but "on average" these are > > > close(?). Does the ratio V(n)/E(n) approach a particular value? I don't > > > know. > > > > > > Brian Chandler > > > http://imaginatorium.org > > > > Maybe n-gonometry would help, towards n-k-hedrometry. > > > > Re "countable uncountable", does it not seem that nested intervals must > > apply to irrationals else it wouldn't? > > Ross, you seem a pleasant enough chap, but your posts always remind me > of that wonderful scene in the Python film "Eric the Viking", in which > Sekine Tsutomu, a Japanese radio personality I believe, gets a bit part > as the slave master on the ship - thrashing his whip he pours out a > torrent of abuse, totally unrelated to the words in the subtitles, and > one of the rowers - says in best Python-cockney, "Well, it wouldn't be > so baaaad, if we knew wot 'e was saying, would it?" > > OK. What is n-gonometry? What is "n-k-hedrometry"? What on earth does > the oxymoron "countable uncountable" have to do with anything I said? > (It's just a remnant of the original bit of nonsense from the > ineducable German.) I can't parse the rest of the sentence, but I have > no idea what it would mean for "nested intervals" to "apply" to > irrationals. (Hang on, though, I vaguely recall that one of your claims > clear enough to be wrong was some confusion about nested intervals. Who > knows...) > > Brian Chandler > http://imaginatorium.org Konnichiwa, Ever seen any Beat Takeshi movies? Basically this notion of n-gonometry, where trigonometry is about parametric equations that describe the evolving coordinates of an unhinged equilateral triangle, is about the use of other regular polygons or for that matter non-equilateral triangles with given pleasing forms, with various descriptions of the stiffness and restitutive forces and relaxative conditions on the angles, modern classical geometry. Then, n-k-hedrology would be about similar notions generalized to various regular and irregular polyhedra, in dimensions upwards of two. About the nested intervals and irrationals, basically the notion is that as the irrationals are not complete as the reals are complete, gapless that the nested intervals argument can not show the irrationals uncountable. Well-order the reals, do nothing they already have been. There are only and everywhere reals between zero and one. Ross
From: MoeBlee on 5 Nov 2006 21:36 Lester Zick wrote: > >I looked over the context three times. If I excluded anything that has > >a material bearing on what I did include and my remark on it, then you > >can mention those exclusions specifically. Posts are not discredited by > >the mere fact that they don't include all previous quoted matter. > >Nobody has an obligation to include all previous quoted matter in every > >one of his or her posts. > > That only makes you a censor not a mathematician. Not including all quoted material in a post is not censorship. And I never claimed to be a mathematician. MoeBlee
From: MoeBlee on 7 Nov 2006 18:46 Lester Zick wrote: > And you're kinda funny in the head, Moe(x). So lessee. I call you > "chickenshit" and you quote me as saying You're "chickenshit" and > claim that misrepresents your position as being "chickenshit" while > deleting what I can point to to justify your "chickenshittedness" then > invite me to revisit and recollect your "chickenshittedness" on the > web? On the whole it's just more evidence that you're "chickenshit". That has real art to it. Obviously, you've quite a flair. Maybe you should think about making a little side money doing freelance writing for greeting card companies. Moeblee
From: Lester Zick on 8 Nov 2006 17:17
On 7 Nov 2006 15:46:18 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> And you're kinda funny in the head, Moe(x). So lessee. I call you >> "chickenshit" and you quote me as saying You're "chickenshit" and >> claim that misrepresents your position as being "chickenshit" while >> deleting what I can point to to justify your "chickenshittedness" then >> invite me to revisit and recollect your "chickenshittedness" on the >> web? On the whole it's just more evidence that you're "chickenshit". > >That has real art to it. Obviously, you've quite a flair. Maybe you >should think about making a little side money doing freelance writing >for greeting card companies. And you should seriously consider not doing mathematics. ~v~~ |