An uncountable countable set
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From: stephen on 14 Oct 2006 21:39 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >> >>> Why even mention the gedanken at all then? >> >> I am not the one who brought it up. I am not even sure >> why people think it has anything to do with set theory. > It doesn't. It's a distinct SEQUENCE of events, not a set without order. Yes, it is a distinct sequence of events. Each ball is involved in exactly two events. Ball #n is added at time -(1/2)^(floor(n/10) > Set theory doesn't apply. It's just another example of set theorists > trying to claim that everything falls under set theory. This experimant > obviously does not. Set theory is incapable of handling the concept of > sequence in a well-defined way over such a set. >> The whole argument is simply that if -(1/2)^floor(n/10) is >> less than zero (the minutes before noon that the ball is added), >> then -(1/2)^n is less than zero (the minutes before noon the >> ball is removed). This really does not rely on set theory. > No, set theory confuses the issue with its concentration on omega. There > is no such distinct size of the finite naturals. The infinite iterations > are all compressed to a point in this experiment, and since those > operations are a combination of additions and subtractions, set > theorists feel entitled to rearrange the events any way that gets them > their magical results. It's pitiful. The argument does not rely on set theory. Every ball that is added before noon is removed before noon. This is a simple consequence of the fact that for every n, if -(1/2)^(floor(n/10)) <0 then -(1/2)^n < 0 You have never presented any sort of refutation for that argument, and instead are just responding with lame insults. Stephen
From: Tony Orlow on 14 Oct 2006 21:59 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Why even mention the gedanken at all then? >>> I am not the one who brought it up. I am not even sure >>> why people think it has anything to do with set theory. > >> It doesn't. It's a distinct SEQUENCE of events, not a set without order. > > Yes, it is a distinct sequence of events. Each ball is involved > in exactly two events. Ball #n is added at time -(1/2)^(floor(n/10) Indeed, and each iteration of the sequence is involved with 10-1 balls. Which logic seems to have more of a flaw, when they conflict? >> Set theory doesn't apply. It's just another example of set theorists >> trying to claim that everything falls under set theory. This experimant >> obviously does not. Set theory is incapable of handling the concept of >> sequence in a well-defined way over such a set. > > >>> The whole argument is simply that if -(1/2)^floor(n/10) is >>> less than zero (the minutes before noon that the ball is added), >>> then -(1/2)^n is less than zero (the minutes before noon the >>> ball is removed). This really does not rely on set theory. > >> No, set theory confuses the issue with its concentration on omega. There >> is no such distinct size of the finite naturals. The infinite iterations >> are all compressed to a point in this experiment, and since those >> operations are a combination of additions and subtractions, set >> theorists feel entitled to rearrange the events any way that gets them >> their magical results. It's pitiful. > > The argument does not rely on set theory. Every ball that > is added before noon is removed before noon. This is a simple > consequence of the fact that for every n, > if -(1/2)^(floor(n/10)) <0 > then > -(1/2)^n < 0 For every "n e N". No, that's not set theory. No way. The concept that everything that went in also came out is based, as WM would put it, on the concept of a completed infinity. The condensation point at "noon" is a Zeno machine, design for the express purpose of producing a paradox. > > You have never presented any sort of refutation for that > argument, and instead are just responding with lame > insults. > > Stephen > > What are you on? I have presented more than one argument showing that what you claim is false. If only one ball can be removed at any time, and the vase becomes empty, then there has to have been a single ball removed which produced that result. Of course, it had ten added before it, so there have to have been -9 balls in the vase before then. That's obviously incorrect. The problem is precisely characterized as an infinite series which diverges. There is no discontinuity at noon, as expressed in the problem itself. That's a transfinitological obfuscation. Have a nice day! Tony
From: Tony Orlow on 14 Oct 2006 22:01 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Han de Bruijn wrote: >>>> Virgil wrote: >>>> >>>>> In article <66a8$452f4298$82a1e228$30886(a)news2.tudelft.nl>, >>>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>>>> >>>>>> Randy Poe wrote about the Balls in a Vase problem: >>>>>> >>>>>>> Tony Orlow wrote: >>>>>>>> Specifically, that for every ball removed, 10 are inserted. >>>>>>> All of which are eventually removed. Every single one. >>>>>> All of which are eventually inserted. Every single one. >>>>> None are reinserted after being removed but,each is removed after having >>>>> been inserted, so that leaves them all outside the vase at noon. >>>> Huh! Then reverse the process: first remove 1, then insert 10. It must >>>> be no problem in your "counter intuitive" mathematics to start with -1 >>>> balls in that vase. >>> Consider this situation: Start with an empty vase. Add a ball at time 5. >>> Remove it at time 6. >>> >>> How you would translate that into Mathematics? >>> >> What happens between times 5 and 6? > > Nothing. > >> Are there other balls involved? > > No. > Hi David! Well, thanks for that elucidating example. It's all blindingly obvious now. :) Tony
From: Tony Orlow on 14 Oct 2006 22:03 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> How about this problem: Start with an empty vase. Add a ball to a vase >>>>> at time 5. Remove it at time 6. How many balls are in the vase at time >>>>> 10? >>>>> >>>>> Is this a nonsensical question? >>>> Not if that's all that happens. However, that doesn't relate to the ruse >>>> in the vase problem under discussion. So, what's your point? >>> Is this a reasonable translation into Mathematics of the above problem? >> I gave you the translation, to the last iteration of which you did not >> respond. > >>> "Let 1 signify that the ball is in the vase. Let 0 signify that it is >>> not. Let A(t) signify the location of the ball at time t. The number of >>> 'balls in the vase' at time t is A(t). Let >>> >>> A(t) = 1 if 5 < t < 6; 0 otherwise. >>> >>> What is A(10)?" >> Think in terms on n, rather than t, and you'll slap yourself awake. > > Sorry, but perhaps I wasn't clear. I stated a problem above in English > with one ball and you agreed it was a sensible problem. Then I asked if > the translation above is a reasonable translation of the one-ball > problem into Mathematics. If you gave your translation of the one-ball > problem, I missed it. Regardless, my question is whether the translation > above is acceptable. So, is the translation above for the one-ball > problem reasonable/acceptable? > Yes, for that particular ball, you have described its state over time. According to your rule, A(10)=0, since 10>6>5. Do go on.
From: Tony Orlow on 14 Oct 2006 22:08
Virgil wrote: > In article <4531023e(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> Where is there defined in the problem any mention of a discontinuity in >> the process? There isn't. > > The discontinuities follow from the statement of the problem. The > "number of balls in the urn" has a discontinuity each time it changes. The formula for the number of balls after each iteration is linear. It does not change direction. While the set may be discrete, the formula mapping the set from the naturals is continuous in the reals. Either you see the significance of this, or you don't, I suppose. >>>> What causes a discontinuity at noon? I'll tell you. The von Neumann >>>> limit ordinals. That's schlock. >>> Um, that's foaming at the mouth. Von Neumann limit ordinals have >>> nothing to do with it - the very simplest notion of the natural numbers >>> being an unending sequence - Wolf Kirchmeir's six? eight?-year old >>> grandson's understanding - is absolutely all that is needed. You can't >>> grasp the notion of an unending sequence, which is why you are in such >>> a total tangle. >> I am in no tangle. That you cannot see that your conclusion is based on >> the notion of omega as the first limit ordinal and first discontinuity >> in the ordinals is disappointing. If the sequence never ends, then noon >> never comes. > > The notion that an infinite sequence cannot take occur within a finite > time interval has been dead since Zeno. > You miss the point, as usual. One can look at this in time, in which case the iterations all become squashed together at noon and indistinguishable. Or, one can look at this in iterations, in which case the sum clearly diverges. The Zeno machine is a deliberate attempt to produce a paradox. Ho hum. > >> The process at noon is not well defined, since the distinction between >> iterations disappears. How do you know there are countably many >> iterations, and not some uncountably number? You don't. You base your >> argument on all iterations being finite, but there is no least upper >> bound to the finites, because there is no least infinite. > > In TO's world, only TO can say what is or is not there, but in ZF and > NBG, there is a least infinite ordinal. ZF and NBG don't handle sequences or their sums, but only unordered sets, so they really have nothing to say on the matter. |