An uncountable countable set
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From: Mike Kelly on 26 Oct 2006 13:47 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: > > stephen(a)nomail.com wrote: > >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >>> imaginatorium(a)despammed.com wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> As each ball n is removed, how many remain? > >>>>>>> 9n. > >>>>>>> > >>>>>>>> Can any be removed and leave an empty vase? > >>>>>>> Not sure what you are asking. > >>>>>> If, for all n e N, n>0, the number of balls remaining after n's removal > >>>>>> is 9n, does there exist any n e N which, after its removal, leaves 0? > >>>>> I don't know what you mean by "after its removal"? > >>>> Oh, I think this is clear, actually. Tony means: is there a ball (call > >>>> it ball P) such that after the removal of ball P, zero balls remain. > >>>> > >>>> The answer is "No", obviously. If there were, it would be a > >>>> contradiction (following the stated rules of the experiment for the > >>>> moment) with the fact that ball P must have a pofnat p written on it, > >>>> and the pofnat 10p (or similar) must be inserted at the moment ball P > >>>> is removed. > >> > >>> I agree. If Tony means is there a ball P, removed at time t_P, such that > >>> the number of balls at time t_P is zero, then the answer is no. After > >>> all, I just agreed that the number of balls at the time when ball n is > >>> removed is 9n, and this is not zero for any n. > >> > >>>> Now to you and me, this is all obvious, and no "problem" whatsoever, > >>>> because if ball P existed it would have to be the "last natural > >>>> number", and there is no last natural number. > >>>> > >>>> Tony has a strange problem with this, causing him to write mangled > >>>> versions of Om mani padme hum, and protest that this is a "Greatest > >>>> natural objection". For some reason he seems to accept that there is no > >>>> greatest natural number, yet feels that appealing to this fact in an > >>>> argument is somehow unfair. > >> > >>> The vase problem violates Tony's mental picture of a vase filling with > >>> water. If we are steadily adding more water than is draining out, how > >>> can all the water go poof at noon? Mental pictures are very useful, but > >>> sometimes you have to modify your mental picture to match the > >>> mathematics. Of course, when doing physics, we modify our mathematics to > >>> match the experiment, but the vase problem originates in mathematics > >>> land, so you should modify your mental picture to match the mathematics. > >> > >> As someone else has pointed out, the "balls" and "vase" > >> are just an attempt to make this sound like a physical problem, > >> which it clearly is not, because you cannot physically move > >> an infinite number of balls in a finite time. It is just > >> a distraction. As you say, the problem originates in mathematics. > >> Any attempt to impose physical constraints on inherently unphysical > >> problem is just silly. > >> > >> The problem could have been worded as follows: > >> > >> Let IN = { n | -1/(2^floor(n/10) < 0 } > >> Let OUT = { n | -1/(2^n) } > >> > >> What is | IN - OUT | ? > >> > >> But that would not cause any fuss at all. > >> > >> Stephen > >> > > > It would still be inductively provable in my system that IN=OUT*10. > > So you actually think that there exists an integer n such that > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 > ? log 10*BigUn ? -- mike.
From: Tony Orlow on 26 Oct 2006 14:09 Mike Kelly wrote: > Tony Orlow wrote: >> Mike Kelly wrote: >>> Randy Poe wrote: >>>> Tony Orlow wrote: >>>>> Virgil wrote: >>>>>> In article <453e4a85(a)news2.lightlink.com>, >>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> >>>>>>> David Marcus wrote: >>>>>>>> Tony Orlow wrote: >>>>>>>>> David Marcus wrote: >>>>>>>>>> Tony Orlow wrote: >>>>>>>>>>> David Marcus wrote: >>>>>>>>>>>> Tony Orlow wrote: >>>>>>>>>>>>> Your examples of the circle and rectangle are good. Neither has a >>>>>>>>>>>>> height >>>>>>>>>>>>> outside of its x range. The height of the circle is 0 at x=-1 and x=1, >>>>>>>>>>>>> because the circle actually exists there. To ask about its height at >>>>>>>>>>>>> x=9 >>>>>>>>>>>>> is like asking how the air quality was on the 85th floor of the World >>>>>>>>>>>>> Trade Center yesterday. Similarly, it makes little sense to ask what >>>>>>>>>>>>> happens at noon. There is no vase at noon. >>>>>>>>>>>> Do you really mean to say that there is no vase at noon or do you mean >>>>>>>>>>>> to say that the vase is not empty at noon? >>>>>>>>>>> If noon exists at all, the vase is not empty. All finite naturals will >>>>>>>>>>> have been removed, but an infinite number of infinitely-numbered balls >>>>>>>>>>> will remain. >>>>>>>>>> "If noon exists at all"? How do we decide? >>>>>>>>>> >>>>>>>>> We decide on the basis of whether 1/n=0. Is that possible for n in N? >>>>>>>>> Hmmmm......nope. >>>>>>>> So, noon doesn't exist. And, there is no vase at noon. I thought you >>>>>>>> were saying the vase contains an infinite number of balls at noon. >>>>>>>> >>>>>>> If the vase exists at noon, then it has an uncountable number of balls >>>>>>> labeled with infinite values. But, no infinite values are allowed i the >>>>>>> experiment, so this cannot happen, and noon is excluded. >>>>>> So did the North Koreans nuke the vase before noon? >>>>>> >>>>>> The only relevant issue is whether according to the rules set up in the >>>>>> problem, is each ball inserted before noon also removed before noon?" >>>>>> >>>>>> An affirmative confirms that the vase is empty at noon. >>>>>> A negative directly violates the conditions of the problem. >>>>>> >>>>>> How does TO answer? >>>>> You can repeat the same inane nonsense 25 more times, if you want. I >>>>> already answered the question. It's not my problem that you can't >>>>> understand it. >>>> Your response requires that the vase contains balls which were >>>> never, by the stated rules, put in. >>>> >>>> You keep saying things like "if the clock runs till noon there are >>>> balls with infinite numbers on them" even though the rules say there >>>> are >>>> no balls with infinite numbers on them. How do you reconcile that? >>>> >>>> If I put in balls 1, 2, 3 and stop, can the clock tick till noon >>>> without >>>> requiring a 4th ball? >>>> >>>> If I specify times for balls 1-1000 only, can the clock till noon >>>> without >>>> requiring a 1001-th ball? >>>> >>>> How is it, in your world, that when I specify times for all natural >>>> numbered >>>> balls, I am required to put in balls that don't have natural numbers? >>> The problem is that Tony thinks time is a function of the number of >>> insertions you've gone through. In order to "get to" any particular >>> time you have to perform the insertions "up to" that point. He then >>> thinks that if you want to "get to" noon, you have to have performed >>> some "infinite" (whatever that means) iterations, where balls without >>> natural numbers are inserted. That this is obviously not what the >>> problem statement says doesn't seem to bother him. Nor that it's >>> absolutely nothing like an intuitive picture of what time is. >> Time is ultimately irrelevant in this gedanken, but if it is to be >> considered, the constraints regarding time cannot be ignored. > > Is time relevent to the question or isn't it? If it isn't, why must > these "constraints" be respected? > >> Events occurring in time must occupy at least one moment. > > I have no idea what this is supposed to mean. > >>> Obviously, time is an independent variable in this experiment and the >>> insertion or removal or location of balls is a function of time. That's >>> what the problem statement says: we have this thing called "time" which >>> is a real number and it "goes from" before noon to after noon and, at >>> certain specified times, things happen. There are only >>> naturally-numbered balls inserted and removed, always before noon. >>> Every ball is removed before noon. Therefore, the vase is empty. >> No, you have the concept of the independent variable bent. > > No, you have the concept of the independent variable bent. > >> The number of balls is related to the time by a formula which works in both directions. > > For any iteration in the sequence of insertions/removals you can work > out what time it occurs at if you know what number the iteration is > indexed by. This doesn't imply that noon "does not exist" unless there > is an iteration that corresponds to it. That is a complete non sequitur > and, I think, the root logical error that you make. > >> So, when does the vase become empty? Nothing can occur at noon, as far >> as ball removals. AT every time before noon, balls are in the vase. So, >> when does the vase become empty > > At every time before noon (after 1 minute to noon) there are balls in > the vase. At noon, there are no balls in the vase. So I guess one would > say the vase "becomes" empty at noon. > >> , and how? > > By every ball that was inserted having been removed. > > Now correct me if I'm wrong, but I think you agreed that every > "specific" ball has been removed before noon. And indeed the problem > statement doesn't mention any "non-specific" balls, so it seems that > the vase must be empty. However, you believe that in order to "reach > noon" one must have iterations where "non specific" balls without > natural numbers are inserted into the vase and thus, if the problem > makes sense and "noon" is meaningful, the vase is non-empty at noon. Is > this a fair summary of your position? > > If so, I'd like to make clear that I have no idea in the world why you > hold such a notion. It seems utterly illogical to me and it baffles me > why you hold to it so doggedly. So, I'd like to try and understand why > you think that it is the case. If you can explain it cog
From: Tony Orlow on 26 Oct 2006 14:10 Randy Poe wrote: > Tony Orlow wrote: >> t=0 is precluded by n e N and t(n) = -1/n. > > Really? > > I hope you will accept as true that noon occurred yesterday. > > Let's define noon yesterday as t=0. Now let's define a set of values > t_n = -1/n seconds for n=1, 2, 3, ... , that is, for all FINITE > natural numbers n. > > Has my giving these names to those times somehow > precluded noon yesterday from occurring? Retroactively? > > - Randy > Do you live in the gedanken? Oy. Nothing happens at noon. Your desired result does not happen before noon. Go back to yesterday and start over.
From: Tony Orlow on 26 Oct 2006 14:11 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>>> imaginatorium(a)despammed.com wrote: >>>>> David Marcus wrote: >>>>>> Tony Orlow wrote: >>>>>>> David Marcus wrote: >>>>>>>> Tony Orlow wrote: >>>>>>>>> As each ball n is removed, how many remain? >>>>>>>> 9n. >>>>>>>> >>>>>>>>> Can any be removed and leave an empty vase? >>>>>>>> Not sure what you are asking. >>>>>>> If, for all n e N, n>0, the number of balls remaining after n's removal >>>>>>> is 9n, does there exist any n e N which, after its removal, leaves 0? >>>>>> I don't know what you mean by "after its removal"? >>>>> Oh, I think this is clear, actually. Tony means: is there a ball (call >>>>> it ball P) such that after the removal of ball P, zero balls remain. >>>>> >>>>> The answer is "No", obviously. If there were, it would be a >>>>> contradiction (following the stated rules of the experiment for the >>>>> moment) with the fact that ball P must have a pofnat p written on it, >>>>> and the pofnat 10p (or similar) must be inserted at the moment ball P >>>>> is removed. >>>> I agree. If Tony means is there a ball P, removed at time t_P, such that >>>> the number of balls at time t_P is zero, then the answer is no. After >>>> all, I just agreed that the number of balls at the time when ball n is >>>> removed is 9n, and this is not zero for any n. >>>>> Now to you and me, this is all obvious, and no "problem" whatsoever, >>>>> because if ball P existed it would have to be the "last natural >>>>> number", and there is no last natural number. >>>>> >>>>> Tony has a strange problem with this, causing him to write mangled >>>>> versions of Om mani padme hum, and protest that this is a "Greatest >>>>> natural objection". For some reason he seems to accept that there is no >>>>> greatest natural number, yet feels that appealing to this fact in an >>>>> argument is somehow unfair. >>>> The vase problem violates Tony's mental picture of a vase filling with >>>> water. If we are steadily adding more water than is draining out, how >>>> can all the water go poof at noon? Mental pictures are very useful, but >>>> sometimes you have to modify your mental picture to match the >>>> mathematics. Of course, when doing physics, we modify our mathematics to >>>> match the experiment, but the vase problem originates in mathematics >>>> land, so you should modify your mental picture to match the mathematics. >>> As someone else has pointed out, the "balls" and "vase" >>> are just an attempt to make this sound like a physical problem, >>> which it clearly is not, because you cannot physically move >>> an infinite number of balls in a finite time. It is just >>> a distraction. As you say, the problem originates in mathematics. >>> Any attempt to impose physical constraints on inherently unphysical >>> problem is just silly. >>> >>> The problem could have been worded as follows: >>> >>> Let IN = { n | -1/(2^floor(n/10) < 0 } >>> Let OUT = { n | -1/(2^n) } >>> >>> What is | IN - OUT | ? >>> >>> But that would not cause any fuss at all. >>> >>> Stephen >>> > >> It would still be inductively provable in my system that IN=OUT*10. > > So you actually think that there exists an integer n such that > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 > ? > > What might that integer be? > > Stephen > How do you glean that from what I said? Your "largest finite" arguments are very boring.
From: MoeBlee on 26 Oct 2006 14:37
Ross A. Finlayson wrote: > Please identify something you see as incorrect or don't understand. What's the point? People have been pointing out your incorrect statements for years. You just sail right past every time. So really think you are correct and that you make sense? It must be nice. MoeBlee |