An uncountable countable set
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From: stephen on 27 Oct 2006 01:15 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. What are you talking about? I defined two sets. There are no balls or vases. There are simply the two sets IN = { n | -1/(2^floor(n/10)) < 0 } OUT = { n | -1/(2^n) < 0 } >> >> So apparently you do not think such an n exists, yet you >> think there are elements in IN that are not in OUT. >> Those are contradictory positions. > 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. But OUT is not a proper subset of IN, unless you believe that there exists an n such that -1/(2^floor(n/10)) < 0 but -1/(2^n) >= 0 If you claim that OUT is a proper subset of IN, you must be able to identify an element that is in IN that is not in OUT. Stephen
From: David Marcus on 27 Oct 2006 01:27 Ross A. Finlayson wrote: > David Marcus wrote: > > > > When mathematicians talk, they know which words have technical meanings > > and which don't. Ross simply uses the words without knowing the > > technical meanings. So, it gives the appearance of mathematics, but > > there is no actual communication of mathematical ideas. > > > > I once tried to teach some mathematics to some friends who weren't > > mathematicians via a weekly lunch seminar. We took a (fairly advanced) > > math book, and started reading it. When I read a math book, I can > > immediately categorize each sentence as definition, theorem, proof, or > > remark, even if the sentence isn't labeled as such. I was a bit > > surprised to discover that my friends weren't picking up on this at all. > > As such, they couldn't even begin to follow the book, since they > > couldn't tell what the purpose of each sentence was in the logical flow. > > They weren't even aware of the convention that a word in italics means > > the sentence is a definition of the word. > > I dispute that. You dispute what, exactly? > Let's see, the nearest mathematics book is a Dover reprint of Meyer's > _Introduction to Mathematical Fluid Dynamics_. So, explain fluid > mechanics. > > Perhaps you feel more secure in your grasp of the subject talking about > set theory? So do I. -- David Marcus
From: David Marcus on 27 Oct 2006 01:32 imaginatorium(a)despammed.com wrote: > > Virgil wrote: > > In article <45417528$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > <snip> > > > > 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. :) > > > > Coming from TO it damns Ross. > > Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this "damns" Robinson's non-standard analysis, does it? Virgil said "Ross", not "Robinson", I believe. -- David Marcus
From: David Marcus on 27 Oct 2006 01:37 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: > >>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> stephen(a)nomail.com wrote: > >>>>>> 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. > > What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets > > IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } It is interesting that when we try to ask Tony a question that doesn't mention balls or vases or time, his answer involves balls, vases, and time. I'm afraid to ask what 1 + 1 is because the answer might be "noon doesn't exist". -- David Marcus
From: Virgil on 27 Oct 2006 02:34
In article <1161925496.451489.322630(a)h48g2000cwc.googlegroups.com>, imaginatorium(a)despammed.com wrote: > Virgil wrote: > > In article <45417528$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > <snip> > > > > 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. :) > > > > Coming from TO it damns Ross. > > Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this "damns" Robinson's non-standard analysis, does it? You are mixing up Ross A. Finlayson with Abraham Robinson. I have nothing against Robinson or his non-standard analysis, in fact, I rather like it. |