An uncountable countable set
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From: Tony Orlow on 17 Oct 2006 22:35 imaginatorium(a)despammed.com wrote: > Lester Zick wrote: >> On 17 Oct 2006 10:37:08 -0700, imaginatorium(a)despammed.com wrote: >> >>> Tony Orlow wrote: >> [. . .] >> >>>> Oh. What was Aleph_0 again? >>> You have, I'm sure been told dozens, if not hundreds of times - Aleph_0 >>> is the name for the cardinality one might explain to children as "you >>> can count, and reach any of them, but the counting never stops". >> So now aleph is an empirical concept, Brian? > > No, Lester. Technically (I expect you'll enjoy hearing that word!), > aleph is a letter in the Hebrew alphabet. At least I believe so - I do > remember John Conway using beth in a lecture, and commenting that it's > a pity people don't know more than one letter of Hebrew. aleph_ beth, gimmel. So? Where are your infinite gimmels, or did Cantor think that sounded too stupid? So, aleph_0 is just a letter with a number sort of tacked on. That's very meaningful. > >> ... I mean unless you >> consider children too lazy or stupid to intuit the subtle virtue of >> set theoretic assumptions underlying their ability to count one by >> one. > > Sentence is a bit long for me to grasp in one go, but no, I don't > consider children too lazy or stupid for anything, in general. That > generally happens when we get older. Indeed. > > Brian Chandler > http://imaginatorium.org >
From: Tony Orlow on 17 Oct 2006 22:37 Lester Zick wrote: > On 17 Oct 2006 12:02:36 -0700, imaginatorium(a)despammed.com wrote: > >> Lester Zick wrote: >>> On 17 Oct 2006 10:37:08 -0700, imaginatorium(a)despammed.com wrote: >>> >>>> Tony Orlow wrote: >>> [. . .] >>> >>>>> Oh. What was Aleph_0 again? >>>> You have, I'm sure been told dozens, if not hundreds of times - Aleph_0 >>>> is the name for the cardinality one might explain to children as "you >>>> can count, and reach any of them, but the counting never stops". >>> So now aleph is an empirical concept, Brian? >> No, Lester. Technically (I expect you'll enjoy hearing that word!), > > Well TECHNICALLY, Brian, any concept you resort to empirical > validation for is an empirical concept. Children learning to count is > scarcely a mathematical validation. Just calling it a cardinality is > no justification whatsoever since I rather doubt children know what > cardinality means. > >> aleph is a letter in the Hebrew alphabet. At least I believe so - I do >> remember John Conway using beth in a lecture, and commenting that it's >> a pity people don't know more than one letter of Hebrew. > > I suspect you mean it's a pity modern mathematikers don't. > >>> ... I mean unless you >>> consider children too lazy or stupid to intuit the subtle virtue of >>> set theoretic assumptions underlying their ability to count one by >>> one. >> Sentence is a bit long for me to grasp in one go, but no, I don't >> consider children too lazy or stupid for anything, in general. That >> generally happens when we get older. > > And for others it begins when children learn that aleph is their > raison d'etre for learning to count. > > ~v~~ Hey, in Israel, do they have alephbeth soup and sing their aleph-beth-gimmels? "...next time won't you....uh...what rhymes with gimmels?" Must be plenty of Hebrew words that do....
From: Tony Orlow on 17 Oct 2006 22:39 Virgil wrote: > In article <26453$4534c7d5$82a1e228$20375(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >> Virgil wrote: >> >>> In article <45341a3a$1(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Well, I think that, while the empty set may easily be taken to represent >>>> 0, 1 is not the set containing 0. That doesn't seem, even at first >>>> glance, like a very accurate model of what 1 is. >>> If TO is not happy with the set representing 1 containing a single item >>> does TO want the set representing 1 to contain more or less that single >>> item? >> That single item is the EMPTY set, pasted between curly braces. > > HdB is missing my point here. > > If TO accepts {} as representing 0 but does not like {{}} as > representing 1, what does TO suggest replace {{}} as representing1? It's a set that contains the empty set. I don't see that as signifying very much, or being particularly useful.
From: Tony Orlow on 17 Oct 2006 22:41 Virgil wrote: > In article <26453$4534c7d5$82a1e228$20375(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >> Virgil wrote: >> >>> In article <45341a3a$1(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Well, I think that, while the empty set may easily be taken to represent >>>> 0, 1 is not the set containing 0. That doesn't seem, even at first >>>> glance, like a very accurate model of what 1 is. >>> If TO is not happy with the set representing 1 containing a single item >>> does TO want the set representing 1 to contain more or less that single >>> item? >> That single item is the EMPTY set, pasted between curly braces. > > HdB is missing my point here. > > If TO accepts {} as representing 0 but does not like {{}} as > representing 1, what does TO suggest replace {{}} as representing1? 1 represents the finite unit. 1 is arbitrary in that respect. 1 is 0+1. It marks the end of an interval containing infinitely many reals, not a container for a container full of nothing.
From: Tony Orlow on 17 Oct 2006 22:53
cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>>> cbrown(a)cbrownsystems.com wrote: >>>>> Tony Orlow wrote: >>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>>>> Tony Orlow wrote: >>>>>>>>>> cbrown(a)cbrownsystems.com wrote: >>>>> <snip> > > <snipitty-snip> > >>> Do you accept the above statements, or do you still claim that there is >>> /no/ valid proof that ball 15 is not in the vase at t=0? >>> >> 15 is a specific finite number for which we can state its times of entry >> and exit. > > Agreed, > >> At its time of exit, balls 16 through 150 reside in the vase. > > Agreed. > >> For every finite n in N, upon its removal, 9n balls remain. > > "upon its removal" = "at the time of ball n's removal"; Agreed. > >> For every n >> e N, there is a finite nonzero number of balls in the vase. > > "For every n e N, there is a finite non-zero number of balls in the > vase at t = -1/n". Agreed. > >> Every >> iteration in the sequence is indexed with an n in N. > > "Balls are only added or removed at a time t = -1/n for some natual n." > Agreed. > >> Therefore, nowhere >> in the sequence... > > ..., i.e, at no time t such that t = -1/n for some natural n, ... > >> is there anything other than a finite nonzero number of >> balls in the vase. > > Agreed. > >> Now, where, specifically, in the fallacy in that argument? >> > > Well, what do you state is the conclusion of this argument? You have agreed with everything so far. At every point before noon balls remain. You claim nothing changes at noon. Is there something between noon and "before noon", when those balls disappeared? If not, then they must still be in there. > > If the conclusion of this argument is "we cannot therefore state that > ball 15 is not in the vase at t=0", I really don't see how you have > addressed the issue. You agree that ball 15 is removed, and not put > back in the vase at any time before or at noon; and I think you would > agree that if if a ball is not put in the vase, it cannot be in the > vase. Therefore ball 15 is not in the vase at noon; and nothing you > said above challenges the logic of this conclusion. Of course not. Ball 15 is gone. > > If your conclusion is "therefore, at t=0, there must be a finite > nonzero number of balls in the vase", then the fallacy is called non > sequituur - it doesn't logically follow. There must be a nonzero number, unless soemthing occured between "before noon" and noon. Is there something between those two? > > * Because t=0 is /not/ a time such that t = -1/n for some natural n; Is there something between x=0 and x<0? > > * Therefore your statements regarding exclusively times t that /are/ of > the form -1/n for some natural n do not /automatically/ apply to t=0. > > In order to make a conclusion about t=0 from your statements, you must > appeal to some /other principle/ which connects the state of the vase > at times t = -1/n for natural number n; with the state of the vase at > times which are /not/ of the form t = -1/n for some natural number n; > such as t=-2/3 or (more saliently) t=0. Which I have done. You claim a change of state between "x<0" and "x=0". What is between 0 and less than 0? > >>>> Your statement concerning n does >>>> not cover noon, because noon=f(oo), and oo is outside your range. >>> You've lost me. >> Nothing happens at noon, if all sequential iterations are finite, given >> the time sequence. > > That is not inconsistent with the statement "ball 15 is not in the > vase at noon." > What insoncistency do you perceive? Is "being in" or "being out" something that happens/changes, or just a state of affairs at that time? >> At all moments before noon, as has been conceded, >> there are a nonzero number of balls in the vase. >> > > That is not inconsistent with the statement "ball 15 is not in the > vase at noon." WHAT??? You agree with both statements, so you are accusing yourself of inconsistency. Reread. > >>> What is f? What does it mean to say "noon = f(oo)"? How does this >>> disprove the assertion that ball 15 is not in the vase at t=0? >>> >> It means that every finite iteration occurs before noon, so the only >> ones that can happen AT noon are infinite. > > The only "iterations" that occur are associated with natural numbers; > there are no "infinite iterations" at all that "happen" at any time. So, in the experiment, nothing happens at noon. > >> You have no infinite >> iterations... > > ... just as I have no solid gold statuettes of Richard Nixon kissing > Henry Kissinger... Oh, shut up. > >> ... so noon does not occur, > > What on earth does that even mean? Where, in the problem statement, do > we conclude that "t=0" is not something that can "occur"? Which values > of t can "occur"? Those associated with finite n. ><snip endless obnoxiousness> |