An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: cbrown on 27 Oct 2006 00:34 imaginatorium(a)despammed.com wrote: > MoeBlee wrote: > > Lester Zick wrote > > > > even more sheer brilliance! > > <...> > > > I won't quote more that opening and close, as one can become > > overwhelmed by so much wisdom from just one man in just one day. > > Ah, I see you've noticed. Meanwhile, "Have you tried searching the > archive for Zick + transcendental?" > > http://groups.google.com/group/sci.math/search?group=sci.math&q=zick+transcendental&qt_g=1&searchnow=Search+this+group > I think we have a decent chance of monopolizing the "zick + chaotic" namespace. Are we on? Cheers - Chas
From: cbrown on 27 Oct 2006 00:39 Ross A. Finlayson wrote: > Tony Orlow wrote: > > > > > For what it's worth, and I know this doesn't add a lot of credibility to > > Ross in your eyes, coming from me, but I think Ross has a genuine > > intuition that isn't far off with respect to what's controversial in > > modern math. Sure, he gets repetitive and I don't agree with everything > > he says, but his cryptic "Well order the reals", which I actually > > haven't seen too much of lately, is a direct reference to his EF > > (Equivalence Function, yes?) between the naturals and the reals in > > [0,1). The reals viewed as discrete infinitesimals map to the > > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > > answer your question, I think Ross makes some sense. But, of course, > > coming from me, that probably doesn't mean much. :) > > > > TOE-Knee > > Hi, > > What is this IFR, "inverse function rule"? I've heard you mention it. That which can be done; can be undone. Cheers - Chas
From: imaginatorium on 27 Oct 2006 00:40 David Marcus wrote: > stephen(a)nomail.com wrote: > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > stephen(a)nomail.com wrote: <snip> > > >> The problem could have been worded as follows: > > >> > > >> Let IN = { n | -1/(2^floor(n/10) < 0 } > > >> Let OUT = { n | -1/(2^n) } > > > > > I think you meant > > > > > Let OUT = { n | -1/(2^n) < 0 } > > > > >> What is | IN - OUT | ? > > >> > > >> But that would not cause any fuss at all. > > > > > I wonder. Does anyone reading this think | IN - OUT | <> 0? > > > > Tony does. > > Apparently so. I wonder if he would have thought that if he hadn't first > read the balls and vase problem. I don't remember offhand, but Tony's debut may have been triggered by the round of ballovasia a year or so ago. But in any event, there was a long, long, period in which we got the Orlovian version of "mathematics": a central feature of which is that if you take a "the set of all natural numbers", and make one "the set of all even naturals" from it by chopping out the odd numbers, then you make another "the set of all even naturals" from by multiplying by 2 the members of the "the set of all natural numbers" you started with, then you get two different "the set(s) of all even naturals", one being twice as numerous as the other. Tony seems to respond to the obvious problems of this multiplication of entities by "declaring" this and that, here and there, so he always gets the answer his razor-sharp intuition tells him is the one he was looking for. Well, sometimes it's entertaining - been rather repetitive of late though. Brian Chandler http://imaginatorium.org
From: Virgil on 27 Oct 2006 00:41 In article <4541825d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > So at least one of the assumptions (1)..(8) and (T1) must be discarded > > if we are to resolve this. What do you suggest? Which of (1)..(8) do > > you want discard to maintain (T1)? > > > > Cheers - Chas > > > > This is a very good question, Chas. Thanks. I'll have to think about it, > and I'm rather tired right now, but at first glance it seems like it > could be a sound analysis. I've cut and pasted for perusal when I'm > sharper tomorrow. > > Cheers - Tony Sharper than what?
From: Virgil on 27 Oct 2006 00:43
In article <454184bc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > > > How do I glean that? You claim that IN does not equal OUT. > > IN contains all n such that > > -1/(2^floor(n/10)) < 0 > > and OUT contains all n such that > > -1/(2^n) < 0 > > > > Your claim that IN=OUT*10 (I am guessing you meant |IN|=|OUT|*10), > > so presumably IN is bigger than OUT, and IN contains elements > > that are not in OUT. The only way n can be an element of IN, > > but not of out is if > > -1/(2^floor(n/10)) < 0 > > but > > -1/(2^n) >= 0 > > Incorrect. For every n, finite or infinite, when the nth ball is > removed, 9n remain. Either you get to t=0, in which case all finite > balls are indeed gone, but replaced by uncountably many infinite balls, > or you don't get to noon. Until TO can understand the Hilbert Hotel, he will never understand why removing all the balls leaves the vase empty. > No, it is the only conclusion consistent with the notion that a proper > subset is alway smaller than the superset. It is contradictory to the > notion that a simple bijection indicates equal size for infinite sets. > The mapping formulas must be taken into account for proper comparison in > that case. The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? |