An uncountable countable set
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From: Virgil on 15 Oct 2006 00:23 In article <45319e2d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <453108b5(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >>> So how many balls are left in the vase at 1:00pm? > >>> > >> If you paid attention to the various subthreads, you'd know I just > >> answered that. Where the insertions and removals are so decoupled, there > >> is no problem. Where the removal of a ball is immediately preceded and > >> succeeded by insertions of 10, the vase never empties. > > > > It may that it never "empties", but at noon, and thereafter, it has > > "become empty". > > > > And when did that happen? At the invisible transition from forenoon to noon. > > > > > The only necessary constraint on insertions of balls into the vase and > > removals of balls from the vase is that each ball that is to be removed > > must be inserted before it can be removed, and, subject only to that > > constraint, the set of balls remaining in the vase at the end of all > > removals is independent of both the times of insertion and of the times > > of removal. > > > > To argue otherwise is to misrepresent the problem. > > You already said that.....WRONG!!!! What is wrong about it? > > There is the additional constraint that, before removing any ball, ten > have been inserted. Then let us put all the balls in at once before the first is removed and then remove them according to the original time schedule. Does TO claim that by putting balls in earlier there can be at ANY time fewer balls in the vase that when putting them in later? But when one puts them all in early enough, it becomes obvious that the vase must be empty at noon. So TO must argue that having more balls in the vase at all times before noon results in less balls in the vase at noon. Now that is ...REALLY... WRONG!!! This makes it impossible for the number of balls to > ever decrease, except by that one, before increasing, and for it ever to > "become" empty.
From: imaginatorium on 15 Oct 2006 07:03 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> imaginatorium(a)despammed.com wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: <snipple-snapple> > >>> Consider the function step0: R -> R mapping x to 0 if x<0 and mapping x > >>> to 1 if x>=0. > >> A discontinuous function at x=0. <snip> > >> A function with such a declared discontinuity has two limits at that > >> point, depending on the direction of approach. So, what else is new? > > Ah. Is a "declared discontinuity" somehow significantly different from > > a simple discontinuity? I mean, is there such a thing as an "undeclared > > discontinuity" to which different rules apply? (I've no idea: this is > > not normal mathematical terminology you see.) > > You have defined your step function with an explicit discontinuity. > There is no explicit discontinuity in the gedanken under discussion. The > discontinuity is introduced with the application of omega. Of course step0() is a step function, and it has a discontinuity. But all I have done is *define* it, which I can do without any step in the definition called "declaring a discontinuity". In the ball/vase thought experiment, of course there is a discontinuity at noon - even just from the words of the description, it is clear there is an abrupt change from ever-more-frenetic vase filling/emptying to a state of calm where nothing changes. Suppose I define the following function, referring to sliver-1, which is the area between y=-2/x and y=-1/x for x<0. ("sleight" stands for 'sliver height', not 'sleight of hand'...) sleight(x) = -2/x +1/x for x<0; 0 elsewhere I suppose we agree that sleight(), which increases without limit as negative x->0, has a discontinuity at zero. Is this a "declared discontinuity"? <snip> > > According to your "view" then, there is no discontinuity at noon - is > > that right? The number of balls identified by natural numbers increases > > without limit, and despite the fact that there is no ball not removed > > before noon, at ten past an unlimited number of them are somehow still > > lurking in the vase? > > The process at noon is not well defined, On the contrary, the process *at noon* is completely well-defined. A ball is inserted in the vase or removed from the vase only at a time that is -1/n for some pofnat n. There is no pofnat m such that -1/m = 0. Therefore no ball is either inserted or removed at noon. (This really is elementary, you know.) > ... since the distinction between > iterations disappears. How do you know there are countably many > iterations, and not some uncountably number? You don't. Of course I do. The problem explicitly says balls with natural numbers (pofnats) on them. The sequence either of insertions or of removals is immediately mapped onto the pofnats. There can - by definition - only be a countable sequence of pofnats. (Actually, in non-mumbo-jumbo, "uncountable sequence" is a contradiction in terms.) > You base your > argument on all iterations being finite, Yes, because the problem explicitly says that every ball inserted or removed is marked with a pofnat. > ... but there is no least upper > bound to the finites, Congratulations. You got something right. > because there is no least infinite. Uh, no. There is no upper bound to the pofnats (which you so cutely call "the finites"), because they go on for ever. Not because something which would come after them if you went on for a bit more than ever has some property, because it makes no sense to talk of going on for more than ever. > > Look, I know my "sliver" corresponds to a slightly different sequence, > > but it's simpler. Consider the sliver between y=-2/x and y=-1/x, for > > x<0. Consider it "hatched" with horizontal lines on integral values of > > y. Think of every one of the horizontal bars as representing a time > > some ball spends in a vase. You seem to agree that the sliver goes ever > > upward, ever closer but not actually reaching the y-axis. If we were to > > travel upward, we would see each line corresponding to a ball's stay in > > the vase - always in then out halfway towards the y-axis; and > > importantly, this viewing journey would never end. > > So, where is "noon" in your graph? Where x=0. If by "graph" you mean the sliver, then as I have said several times, without your apparent disagreement, this sliver is bounded by two hyperbola lobes, both of which get closer and closer and closer and closer to the y-axis, without ever actually reaching it. (That's why the value of sleight(0) is zero; there is no sliver _on_ the y-axis.) > > But if we were to travel along the x-axis towards the origin, looking > > upwards (this is maths, not physics; we pretend we could view the > > sliver however far away), we would notice that the number of balls was > > increasing without limit. Then we would reach the origin. Looking up we > > would see the sliver to the left of the y-axis. You agreed at one point > > that the sliver is entirely in the neg-x/pos-y quadrant, so obviously > > there is no sliver to the right of the y-axis. > > > > But in your view (do I understand?) somehow there would just be some > > more sliver that had crept around the "top"? Or what? Do enlighten > > us... > > To evaluate the width of the sliver at any point, one needs to specify > that point. If you do not have a variable n, then you do not have an > equation. Your point escapes me here. Is maths restricted to the activity of "having an equation". (I can see that school maths might look that way...) > You throw all the naturals in a bag and pretend you have some > specific number of them, No I do not. It is only _you_ who talks of "specific infinities" and various other nonsense in which you pretend "Big'un" (or whatever it is at the moment) is part of finite arithmetic. > ... but if you try to use your "number" as if it's specific, it breaks. Perhaps you could quote me doing this - I don't recognise it other than as just what you keep doing. Brian Chandler http://imaginatorium.org
From: Han.deBruijn on 15 Oct 2006 14:06 Tony Orlow schreef: > Han de Bruijn wrote: > > > > So the axiom of infinity says that you can get everything from nothing. > > This is contradictory to all laws of physics, where it is said that you > > pay a price for everything. E.g. mass and energy are conserved. > > Han, you can't really be looking for conservation of energy or momentum > or mass in abstract mathematics, can you? This axiom basically defines > the infinite linear inductive set. Given this method of generation, > there should be things we can say about the set, no? So to speak, Tony. In physics and economics, you can't get something for nothing. Nothing just gives nothing. You must have _something_ to start with. I find the idea absurd that natural numbers can be built by putting curly braces around the empty set. Han de Bruijn
From: Han.deBruijn on 15 Oct 2006 14:09 MoeBlee schreef: > Han de Bruijn wrote: > > Virgil wrote: > > > Axiom of infinity: There exists a set x such that the empty set is a > > > member of x and whenever y is in x, so is S(y). > > > > Which is actually the construction of the ordinals. Right? > > Wrong. Don't understand why that's wrong. Please explain. Han de Bruijn
From: Virgil on 15 Oct 2006 14:38
In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > I find the idea absurd that natural numbers can be built > by putting curly braces around the empty set. > > Han de Bruijn It appears as if much of useful mathematics is ultimately based on what HdB finds absurd. |