An uncountable countable set
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From: Tony Orlow on 28 Sep 2006 07:19 Han de Bruijn wrote: > Virgil wrote: > >> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, >> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >> >>> Randy Poe wrote, about the Balls in a Vase problem: >>> >>>> It definitely empties, since every ball you put in is >>>> later taken out. >>> >>> And _that_ individual calls himself a physicist? >> >> Does Han claim that there is any ball put in that is not taken out? > > Nonsense question. Noon doesn't exist in this problem. > > Han de Bruijn > That's the question I am trying to pin down. If noon exists, that's when the vase supposedly empties, since it doesn't do before then. If the limit doesn't "actually occur", then vase never empties (not that it would anyway).
From: Tony Orlow on 28 Sep 2006 07:32 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: > >>> Consider a (notional, theoretical, mathematical, not physical) x-y >>> plane. That is, an area in which there is a point (0,0) in some >>> particular place, an x-axis, y-axis, and points are identified by >>> coordinates x and y, using (in normal maths) real values for these >>> coordinates. Consider (for convenience) that this plane is embedded in >>> a notional graphics application, with a "Fill" function. So if we draw >>> the circle x^2 + y^2 = 49 (centre origin, (constant! Zick, be quiet!) >>> radius 7), then click with the Fill function on the point (2,1), it >>> fills the circle, and no paint spills outside that radius 7. >>> >>> Now suppose we have the graphs of x=2 and x=5. Vertical lines, >>> extending up and down without limit. Suppose we click with the Fill >>> function on the point (3, 4), what would you say happens? Obviously >>> paint fills the vertical strip of width 3. Would you say that any paint >>> was able to "spill" around the (nonexistent!) "top" of either of the >>> graphs, and somehow fill more of the plane than this strip, or would >>> you say we just get a (vertically) unbounded strip of blue? (Goddabe >>> blue!) > >> I'd have to agree that it would fill the strip only. Proceed, but it >> would be nice to know the context of the question. > > Ok, well just as a diversion: suppose you were on > sci.comp.graphics.crank, and one of the residents produced a long, > rambling argument, including mention of Planck's constant, twin-slit > experiments and more, at the end of which was a claim that outside the > strip would also be a very pale (ok "infinitesimally pale"!?) blue. How > would you try to justify your claim that the blue fills the vertically > unbounded strip only? I'd have to see their fill algorithm to see what their malfunction is. > > Note that when discussing the behaviour of a real-world graphics > program, within a bounded window, it's possible to discuss the > paint-filling as a terminating procedure. With an unbounded strip, it > obviously isn't. So I would say something like the following: for the > paint to spill outside the vertical lines bounding the strip, there > must be a path from a point inside to a point outside. But since the > x-coordinate of the points on the path must go from (say) 4 to 6, at > some point it must be 5; and that point must be a point on the > boundary, so it would have crossed the boundary, and it's not allowed > to cross the boundary, so this can't have happened. Any general fill algorithm would probably leave some section of that infinite strip un-blued. > > ---- back to the point ---- > > Now consider some other graphs: > > y=1/x, fill from the point (0, 0) - get blue lower left and upper right > quadrants, plus filling out to the white lobes that almost fill the > upper left and lower right quadrants. OK? (Graph is a hyperbola) > > Now consider the following two hyperbola-halves: > > y1 = -1/x (for negative x) > y2 = -2/x (for negative x) > > Each of these is a lobe in the upper left quadrant, OK? > > Clicking on (-23, 34) would fill just the lobe formed by y2=-2/x (since > this curve is always above and to the left of the other one); clicking > in the lower right quadrant would fill three quadrants, and the area up > to the y1 curve, leaving a (slightly larger) upper left white lobe. (I > hope all this terminology is clear.) > > Would you agree that clicking on (-1, 1.5) fills the sliver between the > two hyperbola lobes? > (I say "sliver", though the area is infinite, since sum(1/n) doesn't > converge - plus a bit of hand-waving.) > > Do you see a connection to the original problem? No. Would you care to be a little more explicit? > > Brian Chandler > http://imaginatorium.org >
From: imaginatorium on 28 Sep 2006 08:19 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Consider a (notional, theoretical, mathematical, not physical) x-y > > plane. That is, an area in which there is a point (0,0) in some > > particular place, an x-axis, y-axis, and points are identified by > > coordinates x and y, using (in normal maths) real values for these > > coordinates. Consider (for convenience) that this plane is embedded in > > a notional graphics application, with a "Fill" function. So if we draw > > the circle x^2 + y^2 = 49 (centre origin, (constant! Zick, be quiet!) > > radius 7), then click with the Fill function on the point (2,1), it > > fills the circle, and no paint spills outside that radius 7. > > > > Now suppose we have the graphs of x=2 and x=5. Vertical lines, > > extending up and down without limit. Suppose we click with the Fill > > function on the point (3, 4), what would you say happens? Obviously > > paint fills the vertical strip of width 3. Would you say that any paint > > was able to "spill" around the (nonexistent!) "top" of either of the > > graphs, and somehow fill more of the plane than this strip, or would > > you say we just get a (vertically) unbounded strip of blue? (Goddabe > > blue!) > I'd have to agree that it would fill the strip only. Proceed, but it > would be nice to know the context of the question. Ok, well just as a diversion: suppose you were on sci.comp.graphics.crank, and one of the residents produced a long, rambling argument, including mention of Planck's constant, twin-slit experiments and more, at the end of which was a claim that outside the strip would also be a very pale (ok "infinitesimally pale"!?) blue. How would you try to justify your claim that the blue fills the vertically unbounded strip only? Note that when discussing the behaviour of a real-world graphics program, within a bounded window, it's possible to discuss the paint-filling as a terminating procedure. With an unbounded strip, it obviously isn't. So I would say something like the following: for the paint to spill outside the vertical lines bounding the strip, there must be a path from a point inside to a point outside. But since the x-coordinate of the points on the path must go from (say) 4 to 6, at some point it must be 5; and that point must be a point on the boundary, so it would have crossed the boundary, and it's not allowed to cross the boundary, so this can't have happened. ---- back to the point ---- Now consider some other graphs: y=1/x, fill from the point (0, 0) - get blue lower left and upper right quadrants, plus filling out to the white lobes that almost fill the upper left and lower right quadrants. OK? (Graph is a hyperbola) Now consider the following two hyperbola-halves: y1 = -1/x (for negative x) y2 = -2/x (for negative x) Each of these is a lobe in the upper left quadrant, OK? Clicking on (-23, 34) would fill just the lobe formed by y2=-2/x (since this curve is always above and to the left of the other one); clicking in the lower right quadrant would fill three quadrants, and the area up to the y1 curve, leaving a (slightly larger) upper left white lobe. (I hope all this terminology is clear.) Would you agree that clicking on (-1, 1.5) fills the sliver between the two hyperbola lobes? (I say "sliver", though the area is infinite, since sum(1/n) doesn't converge - plus a bit of hand-waving.) Do you see a connection to the original problem? Brian Chandler http://imaginatorium.org
From: Dik T. Winter on 28 Sep 2006 09:29 In article <7b55f$451ba2f8$82a1e228$18740(a)news2.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: .... > OK. Virgil corrected this error. But even then. The correct thing would > IMHO be a theorem and not a definition. > > Theorem: 0.33333 .. = 1/3 No. It might be a theorem if 0.333... was defined, so you need a definition first. But the only definition I know of is the one similar to the definition of Virgil. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Randy Poe on 28 Sep 2006 09:41
Tony Orlow wrote: > Randy Poe wrote: > > On rereading this, I think there was some confusion. Whether > > through Tony deliberately misquoting, or a misunderstanding > > on my part or Tony's, I'm not sure which. But Tony seems to > > have conflated a statement I made about emptying with one > > about filling. > > > > Tony Orlow wrote: > >> Randy Poe wrote: > > > > This was about emptying: > > > >>> It definitely empties, since every ball you put in is > >>> later taken out. > >> So, it definitely empties...... > >> > >>>> And, at the same time you > >>>> say it does not do so at noon, nor does it do so before noon. When does > >>>> this occur? > > > > This was about "filling" when I said it: > > > >>> It doesn't. > >> ...but it doesn't! > > > > What I would say about emptying is that the vase is empty > > at noon, but there is no identifiable time before noon at which > > we can say "the last ball was taken out then". > > > > At any time before noon, there are balls in the vase. There > > is no time we can say "there goes the last ball out" since there > > is no last ball in. > > > > If the vase is empty at noon, but not before, how can that not be the > moment that it becomes empty? "Becomes empty" implies a transition from not-empty to empty. What time was it in the moment just before noon? What is the number of the ball which, when removed, makes the vase empty? I know the kind of nonsense you will spout in answer to those questions, but the answers within our axiom system are: (1) there is no t<noon which is the moment just before noon. For any t<noon, there is t < t' < noon. (2) There is no such ball. Here are the Tony gobbledgook answers: (1) noon - 1/oo (2) Ball number omega In TO-matics, one can confidently give an answer like number 2 despite the fact that one can also agree that no ball numbered omega is ever put into the vase. But in mathematics and logic, we don't get to keep a set of self-contradictory assumptions around, only using the ones we want as needed. - Randy |