An uncountable countable set
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From: Tony Orlow on 27 Sep 2006 22:27 Randy Poe wrote: > On rereading this, I think there was some confusion. Whether > through Tony deliberately misquoting, or a misunderstanding > on my part or Tony's, I'm not sure which. But Tony seems to > have conflated a statement I made about emptying with one > about filling. > > Tony Orlow wrote: >> Randy Poe wrote: > > This was about emptying: > >>> It definitely empties, since every ball you put in is >>> later taken out. >> So, it definitely empties...... >> >>>> And, at the same time you >>>> say it does not do so at noon, nor does it do so before noon. When does >>>> this occur? > > This was about "filling" when I said it: > >>> It doesn't. >> ...but it doesn't! > > What I would say about emptying is that the vase is empty > at noon, but there is no identifiable time before noon at which > we can say "the last ball was taken out then". > > At any time before noon, there are balls in the vase. There > is no time we can say "there goes the last ball out" since there > is no last ball in. > > - Randy > If the vase is empty at noon, but not before, how can that not be the moment that it becomes empty?
From: Virgil on 27 Sep 2006 23:30 In article <451b3097(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451a8f41(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >> The question boils down to whether 0^0 is 1. > > > > 0^0 is, in any particular context, what it is defined to be. > > There are contexts in which it is more useful to have it mean 1 and > > others where it is more useful to have it mean 0. > > > > > > But...but...but how can you reconcile those two answers??? :o As they apply in different contexts, no need to reconcile them. > > In which contexts do you find it more convenient for it to be 0? When one wants f(x) = 0^x to be a continuous function for x >=0. > > > > > > >>>> There is confusion about my "definition" of infinitesimals, because I > >>>> can see the validity both in nilpotent infinitesimals and in those that > >>>> are further infinitely divisible. > >>> Until TO can come up with an axiom system which simultaneously allows > >>> his infinitesimals to be both nilpotent and not, he is in trouble. > >>> > >> For purposes of measure on the finite scale, infinitesimals can be > >> considered nilpotent. That's all. Do you disagree? > > > > I disagree that scale changes can convert between zero and non-zero. > > Infinite scale changes can. Not in my book. > > > > > There are approximation methods is which products of small quantities > > are regarded as negligible in comparison to the quantities themselves, > > but they are always just approximations. > > Sure, but how negligible are those products? Negligible is like pregnant in that respect.
From: Virgil on 27 Sep 2006 23:36 In article <451b3296(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > You must have been a strange 10 year old, like that kid > down the block that used to pull the legs off of roaches. Only those that looked like TO. > >> > >>> So the reason I don't say it's full "an infinitesimal time > >>> before noon" or "some other time before noon" is that > >>> I don't say it's full. > >> But, you do say it's full or empty, right? One can easily say that it is empty at any time at which every ball that was put in has been taken out again. Does TO suggest that at any time after noon there is any ball that was put in that was not also taken out? > > > > So your conclusion from my statement that I would never > > say it's full is that sometimes I would say it's full? > > Uh, you would say it contains an infinite number of balls in some > circumstances, as I understand it. Then you misunderstand it. > > If you say it empties, then you would agree that it either fills or it > empties. When does it empty? You say, not before noon. You also say > this does not occur at noon, but after noon there are no balls left. So > when does this occur? When every ball that was put in has also been taken out again.
From: cbrown on 27 Sep 2006 23:36 Tony Orlow wrote: > Virgil wrote: > > In article <451a8f41(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45193e6f(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > >>>> Well, Han, I'm not sure I agree with the statement that reconciliation > >>>> is hopeless. Is it hopeless to reconcile the wave nature of elementary > >>>> entities with their particle nature? > >>> > >>> It is close to hopeless to expect those who reject the law of the > >>> excluded middle (constructionists) and those who insist on it > >>> (formalists) to agree. > >>> > >> If neither can appreciate the other's point, perhaps. Some christians > >> get along quite well with some muslims. > > > > Only by agreeing to disagree. > > Or, by noting the many similarities and few differences between them. > There's not much difference between a good christian and a good muslim. > > >> The question boils down to whether 0^0 is 1. > > > > 0^0 is, in any particular context, what it is defined to be. > > There are contexts in which it is more useful to have it mean 1 and > > others where it is more useful to have it mean 0. > > > > > > But...but...but how can you reconcile those two answers??? :o > > In which contexts do you find it more convenient for it to be 0? > > > > > > >>>> There is confusion about my "definition" of infinitesimals, because I > >>>> can see the validity both in nilpotent infinitesimals and in those that > >>>> are further infinitely divisible. > >>> Until TO can come up with an axiom system which simultaneously allows > >>> his infinitesimals to be both nilpotent and not, he is in trouble. > >>> > >> For purposes of measure on the finite scale, infinitesimals can be > >> considered nilpotent. That's all. Do you disagree? > > > > I disagree that scale changes can convert between zero and non-zero. > > Infinite scale changes can. > > > > > There are approximation methods is which products of small quantities > > are regarded as negligible in comparison to the quantities themselves, > > but they are always just approximations. > > Sure, but how negligible are those products? Like I said, there were > terms in my infinitesimal sections of moving staircase which differed by > a sub-infinitesimal from those in the original staircase. So, they could > be considered to be two infinitesimally different objects in the limit. Here's a thing that confuses me about your use of the term "limit". In the usual sense of the term, every subsequence of a sequence that has as its limit say, X, /also/ has a limit of X. For example, the sequence (1, 1/2, 1/2, 1/3, ..., 1/n, ...) usually is considered to have a limit of 0. And the subsequence (1/2, 1/4, 1/6, ...., 1/(2*n), ...) which is a subsequence of the former sequence has the same limit, 0. But the way you seem to evaluate a limit, the sequence of staircases with step lengths (1, 1/2, 1/3, ..., 1/n, ...) is a staircase with steps size 1/B, where B is unit infinity; but the sequence of staircases with step lengths (1/2, 1/4, 1/6, ..., 1/(2*n), ...), which is a subsequence of the first sequence, would seem to have as its limit a staircase with steps of size 1/(2*B). Unless steps of size 1/B are the same as steps of size 1/(2*B), I don't see how that can be possible. Cheers - Chas
From: Virgil on 27 Sep 2006 23:39
In article <451b3315(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > On rereading this, I think there was some confusion. Whether > > through Tony deliberately misquoting, or a misunderstanding > > on my part or Tony's, I'm not sure which. But Tony seems to > > have conflated a statement I made about emptying with one > > about filling. > > > > Tony Orlow wrote: > >> Randy Poe wrote: > > > > This was about emptying: > > > >>> It definitely empties, since every ball you put in is > >>> later taken out. > >> So, it definitely empties...... > >> > >>>> And, at the same time you > >>>> say it does not do so at noon, nor does it do so before noon. When does > >>>> this occur? > > > > This was about "filling" when I said it: > > > >>> It doesn't. > >> ...but it doesn't! > > > > What I would say about emptying is that the vase is empty > > at noon, but there is no identifiable time before noon at which > > we can say "the last ball was taken out then". > > > > At any time before noon, there are balls in the vase. There > > is no time we can say "there goes the last ball out" since there > > is no last ball in. > > > > - Randy > > > > If the vase is empty at noon, but not before, how can that not be the > moment that it becomes empty? Saying that it is empty is quite different from saying anything about a "last ball". andy does not deny that the vase becomes empty, he just does not say anything about any "last ball out". |