An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Han de Bruijn on 28 Sep 2006 06:15 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: >>>>You stated that you needed counting to determine the successor. That is >>>>false. The successor is defined without any reference to counting. >>> >>>The successor function *is* counting (+1). >> >>Not to those who can't count. Successorship does not require numbers, it >>only requires "next". > > How far would those who cannot count be able to find "the next"? And how do you distinguish "the next" from something previous? This is not a joke. Many young children don't find it trivial that you shouldn't count a thing twice. Or they have forgotten that a thing has been count already and do it for a second time. But, are you really so much smarter than a child? Have _you_ ever tried to count e.g. a hundred marbles in a bag, without having the opportunity to take them out, one by one please? http://hdebruijn.soo.dto.tudelft.nl/fototjes/appels.htm Han de Bruijn
From: Han de Bruijn on 28 Sep 2006 06:24 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > >>In article <1159186907.615747.304410(a)h48g2000cwc.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de wrote: >> >>>1/3 is a number, properly defined, for instance, by the pair of numbers >>>1,3 or 2,6 or 3,9 etc. But 0.333... is not properly defined because you >>>cannot index all positions, you cannot distinguish the positions of >>>this number from those with finite sequences (and you cannot >>>distinguish them from other infinte sequences which could exist, if one >>>could exist). >> >>Def: 0.333... = lim_{n -> oo} Sum_{k = 1..n} 1/3^n > > Definitions (even correct definitions unlike this one) don't guarantee > existence (I used above "to be properly defined" but I meant "to > exist"). Example: The set of all sets is defined but is not existing. OK. Virgil corrected this error. But even then. The correct thing would IMHO be a theorem and not a definition. Theorem: 0.33333 .. = 1/3 Proof: 3 ( 1/10 + (1/10)^2 + (1/10)^3 + ... ) = 3 (1/(1 - 1/10) - 1) = 1/3 : sum of geometric series Han de Bruijn
From: Tony Orlow on 28 Sep 2006 06:54 Virgil wrote: > 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. And, in which contexts would that be desirable? >>> >>>>>> 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. You might want to expand your reading. >>> 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. How mathematical of you.
From: Tony Orlow on 28 Sep 2006 06:57 Virgil wrote: > 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? Yes, at any given time 9/10 of the balls inserted remain. >>> 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. No, your labels misconstrue the problem with your silly fixation on omega. Do I "misunderstand" that if you remove balls 1, then 11, then 21, etc, that the vase will NOT be empty? >> 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. At noon or before noon? You're skirting the issue.
From: Tony Orlow on 28 Sep 2006 07:03
cbrown(a)cbrownsystems.com wrote: > 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 > It's possible because no distinction is currently made between countable infinities, even to the point where a set dense in the reals like the rationals is considered equal to a set sparse in the reals like the naturals. Where there is no parametric understanding of infinity, infinity is just infinity, and 0 is just 0. Where there is a formulaic comparison of infinite sets as n->oo, the distinction can be made. The fact that you have steps of size 1/n as opposed to steps of size 1/(2*n) is a reflection of the fact that the first set has twice the density on the real line as the first. As a proper superset, it SHOULD be larger. So, it's quite possible to make sense of my position, with a modicum of effort. Tony |