An uncountable countable set
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From: Dik T. Winter on 29 Sep 2006 09:44 In article <1159438352.385803.75260(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > > > > > > Let me ask you to, before I can answer such a question. > > > > > > > > What is your definition of "number"? (I asked the same > > > > > > > > from Wolfgang Mueckenheim, but his answer was not > > > > > > > > satisfactory, also not to himself, I think, because he > > > > > > > > never answered to questions about it.) > > > > > > > > > > > > > > Didn't you read my paper on the physical constraints of > > > > > > > numbers? > > > > > > > > > > > > Yes, I read it. Mathematically it makes no sense. > > > > > > > > > > You did not yet recognize it, perhaps later. But you were not > > > > > telling the truth above, were you? > > > > > > > > Where did I not write the truth? > > > > > > Here: > > (See above) > > > > > If I considered Dik as one person, I would write {Dik, Virgil, > > > > > me} for instance. But you are right, there is some ambiguity. > > > > > > > > Yes, so it is not a proper definition. And as such, the above to > > > > me still makes no sense as a proper definition at all. > > > > > > There is a natural number which is the largest one ever mentioned or > > > thought during the lifetime o the universe. It is not properly defined > > > before the universe ceases and probably also not afterwards. But it is > > > or will be (depending on the question of determination or not). > > > Nevertheless, it does not exist yet. We have to live with those > > > improper objects. > > > > So, again, no definition. Where did I not speak the truth? > > Here: "...because he never answered to questions about it". You gave on Usenet a definition of natural number in answer to my question for it. I posted questions about that definition, and you never answered them. So in what way is it a lie when I state that you never answered them? > Most > questions on the representation of a number are answered in my paper. The questions were about the definition you gave in Usenet. And I do not ask about "representations", I ask for a *definition* of the concept "number". A proper, mathematical, definition. > If my > definitions are not proper enough, according to your taste, then > reality is to blame for that. anyhow, your assertion "he never answered > to questions about it" is a lie. It is not, because you never answered my explicit questions about the definition you gave on Usenet. > > > > Oh. So you state. But 1/3 is a number? > > > > > > 1/3 is a number, properly defined, for instance, by the pair of numbers > > > 1,3 or 2,6 or 3,9 etc. But 0.333... is not properly defined because you > > > cannot index all positions, > > > > Again, you *ignore* the definition of that notation as a decimal number. > > I state again, that notation has *no* meaning until some meaning has been > > defined. In mathematics it is defined as the limit of a sequence. If > > you think that definition is invalid, you should seriously consider all > > use of limits in mathematics to be invalid. > > The definition of an object does not provide its existence. Indeed. But when it is *defined* as the limit of a sequence, and if that limit exists, that means that the object does exist. > > The set of known prime numbers is bounded. Period. It is a specific set > > that now and today consists of a fixed number of elements. It may be a > > different set tomorrow, but that is something different, and again, > > tomorrow it will be fixed and bounded. > > The cardinality of the set of prime numbers known today P(t) is as > unbounded as the time variable t of today. If you talk like that you can not talk about "the set of known prime numbers", because that is not a set, but a function of time. For each 't', the outcome is a specific set. A properly defined set does not change over time. So when you talked about the set of known prime numbers, I thought you were talking about the set of prime numbers known at the time you wrote it, as I can give it no other interpretation. > But that does not make time > different. If we find another prime tomorrow, you can be sure that > those primes known today will remain exactly the same. No change. > That's all. Except that you can not talk about a single set of known prime numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: stephen on 29 Sep 2006 10:10 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> 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. >> >> Yes it is a nonsense question, in the sense >> that it is non-physical. You cannot actually perform >> the "experiment". Just as choosing a number uniformly >> from the set of all naturals is a non-physical nonsense >> question. You cannot perform that experiment either. > But you _can_ do it at any time _before_ noon. There is no limit > of the number of balls before noon which converges at noon. > But you _can_ do it with any finite contiguous set of naturals. > Then you find floor(n/a)/n and with limit(n -> oo) find 1/a . > Han de Bruijn But in neither case are you performing the actual "experiment". In the balls in the vase "experiment", for every ball there is a definite time at which it is removed. Your finite approximation throws out that fact, so it is not surprising that it gets the wrong answer. You have fundamentally changed the "experiment". Stephen
From: stephen on 29 Sep 2006 11:46 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> stephen(a)nomail.com wrote: >>> So why is it okay to end up with zero balls, when you never remove >>> any at all, but it is not okay to end up with zero balls when >>> each ball is clearly removed at a definite time? >> Why is it not okay to approach the infinite otherwise than via the limit >> concept? Applied to a _finite_ sequence of events? >> Han de Bruijn > Your first sentence has a double negative in it, so I am > not sure what you intended to say. It is okay to approach > the infinite via the limit concept. Who has ever said otherwise? > Stephen I suppose I should clarify this. You can approach the infinite using the the limit concept, but you always have to be careful when using limits, and you have to be precise about what you mean by the limit. Stephen
From: Tony Orlow on 29 Sep 2006 13:39 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> 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. > > Remember this is not a practical example. It can't be, because no > actual physical computer can simulate a boundless x-y plane. > >>> 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. > > Really? We are talking about the (non-practical, not actually > implementable, [what mathematicians call 'infinite']) notional plane, > and a strip of width 3 that extends indefinitely up and down. An > algorithm by definition terminates, and this strip doesn't, so > obviously no algorithm could fill it. But there doesn't seem to be > anything difficult in painting a strip that goes on forever, other than > that the job can't be completed. Are you suggesting that somehow an > algorithm that purported to paint the whole strip would just stop at > some point? When the strip was a certain height? Even though it painted > without end, it would reach the end? (Oh dear, this all seems rather > familiar...) Yes, your imaginations don't seem to vary much. I am just suggesting that a general-purpose fill algorithm which could fill any finite shape might not work on this shape. If it found a left most point, filled that vertical line up and down until it hit the boundaries of the shape, then went right to the next column, it would never get to that next column, because it would never finish with the first vertical column. If it started by finding a topmost or bottommost point, it would never even get to filling one pixel. Of course you could create an algorithm that worked its way back and forth a row at a time, perhaps going both up and down, and continue filling the shape forever. So, have a nice fantasy about whatever insanity you think I am expressing. (Oh dear, this all seems rather familiar...) > >>> ---- 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 agree, first of all, that there is an area between the two > hyperbola lobes? I'm using this "paint-fill" thing in the hope that it > clarifies what I'm getting at, but it isn't really relevant. Do you > agree that an unbounded line (like the lobe of one hyperbola) divides > this unbounded plane into two areas. Every point (p, q) is either _on_ > the line, or in the upper-left lobe, or in the remaing three > quadrants-plus-the-rounded-bit? Yes, of course. > > Please say you see what I mean, or not. If necessary I can produce some > graphics, but ASCII is a lot easier. > >>> Do you see a connection to the original problem? >> No. Would you care to be a little more explicit? > > OK, suppose you agree that between these two hyperbola we have painted > an unbounded blue sliver, that goes on forever to the left, getting
From: Randy Poe on 29 Sep 2006 13:49
Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > At this point I can hardly remember what we were talking about, but of > course that's partly because you snipped the entire context when you > began this diversion. Randy and I were haggling over whether the vase > empties, and if so, at what exact time? No we weren't. I was making precise statements about the condition of the vase at different times, and you are trying to get me to make vague and incorrect restatements of those precise statements. The vase *is* empty at noon. It *is not* empty at any time before noon. I am refusing to use the verb "empties" because that implies a transition which does not occur. - Randy |