An uncountable countable set
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From: Randy Poe on 30 Oct 2006 22:50 Lester Zick wrote: > It doesn't? My mistake. So there's "no least real" There's no least real. > but a "least integer real"? There's no least integer. There's a least POSITIVE integer. Is there some reason you keep ignoring the critical word POSITIVE? > Hmmm. Curiouser and curiouser. Not at all, when you distinguish between propositions that include that p-word and those that don't. - Randy
From: Virgil on 30 Oct 2006 23:00 In article <MPG.1fb07f27cf5084639897c9(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article <45463580(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> > > wrote: > > > David Marcus wrote: > > > > > > Do you say that g(0) = 0? > > > > > > No, I say that lim(x->0: g(x))=oo. > > > > Does that mean that TO can find any ball inserted before noon which is > > not removed before noon? > > Virgil, the post you are replying to presented a problem without balls > and time. Please don't put balls and time in where they don't belong! Sorry.
From: Virgil on 30 Oct 2006 23:13 In article <4546baa4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow 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: > >>>>> <snip> > >>>>> > >>>>>>> What does that have to do with the sets IN and OUT? IN and OUT are > >>>>>>> the same set. You claimed I was losing the "formulaic relationship" > >>>>>>> between the sets. So I still do not know what you meant by that > >>>>>>> statement. Once again > >>>>>>> IN = { n | -1/(2^(floor(n/10))) < 0 } > >>>>>>> OUT = { n | -1/(2^n) < 0 } > >>>>>>> > >>>>>> I mean the formula relating the number In to the number OUT for any n. > >>>>>> That is given by out(in) = in/10. > >>>>> What number IN? There is one set named IN, and one set named OUT. > >>>>> There is no number IN. I have no idea what you think out(in) is > >>>>> supposed to be. OUT and IN are sets, not functions. > >>>>> > >>>> OH. So, sets don't have sizes which are numbers, at least at particular > >>>> moments. I see.... > >>> If that is what you meant, then you should have said that. > >>> And technically speaking, sets do not have sizes which are numbers, > >>> unless by "size" you mean cardinality, and by "number" you include > >>> transfinite cardinals. > >> So, cardinality is the only definition of set size which you will > >> consider.....your loss. > >> > >>> In any case, it still does not make any sense. I am not sure > >>> what |IN| is for any n. IN is a single set. There is only > >>> one set, and it does not depend on n. In fact, there isn't > >>> an n specified in the problem. Yes I used the letter n in > >>> the set description, but that does not define an entity named 'n'. > >>> > >> There most certainly is an 'n'. The problem describes a repeating > >> process, each repetition of which is indexed with a successive n in n, > >> and during each repetition of which ball n is removed. What do you mean > >> there's no n??? > > > > The ORIGINAL problem. This is a new one, inspired by > > the original, but it is one with no balls, no vases, no > > time steps, no iterations. Just a definition of two subsets > > of the natural numbers, one called IN and one called > > OUT. > > > > The definition of the set IN does not include a definition > > of something called IN(n). > > > > You are being asked to characterize these two subsets. > > > > - Randy > > > > Set-theoretically, they are the same set, by the axiom of > extensionality. That doesn't mean the axioms of ZFC account for all the > information in the problem. There is the combining of +10 and -1 in each > iteration n at time t=-1/n, a coupling that is not being addressed by > your method. You are considering the two sets statically, outside of > time, as completed, but for any n, the max of in is 10 times the max of > out. Since there is nothing in any definition of any set or set theory that I am aware of that allows for non-static sets, the only reasonable way to have sets of differing membership at different times time is by having a function defined over time whose values are different sets at those different times. > Just because these both have some limit at oo, even though they > don't reach it, doesn't mean they reach it at the same time. So what TO must be speaking about here are not sets but functions of time whose values are different sets at different times prior to noon, but which coincide at noon. > They DON'T > reach it, and if they did, if noon occurred, in would reach it in 1/10 > the time as out. TO seems to have entirely lost touch with reality if he thinks that all of the balls could have entered the vase at any time before noon or that any of them can be left at noon. > But,there is no such ending to the finites, and so the > set-theoretic approach using the set N is bogus at its core. What are totally bogus here are TO's puerile objections to perfectly clear logic because he doesn't like the conclusions that logic dictates.
From: imaginatorium on 30 Oct 2006 23:16 Virgil wrote: > In article <45462ba0(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: <snipola> > > >> OH. So, sets don't have sizes which are numbers, at least at particular > > >> moments. I see.... > > > > > > If that is what you meant, then you should have said that. > > > And technically speaking, sets do not have sizes which are numbers, > > > unless by "size" you mean cardinality, and by "number" you include > > > transfinite cardinals. > > > > So, cardinality is the only definition of set size which you will > > consider.....your loss. > > It is the only definition of set size that is known to produce a valid > partial ordering on sets. Huh? I thought cardinality produced a valid *total* ordering on sets. As I have pointed out when all this started, of course everyone knows that the subset relation produces a perfectly valid *partial* ordering on sets. Brian Chandler http://imaginatorium.org
From: stephen on 30 Oct 2006 23:14
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: >>>>> imaginatorium(a)despammed.com wrote: >>>>>> Tony Orlow wrote: >>>> <snip> >>>> >>>>>>> The formulaic relationship is lost in that statement. When you state the >>>>>>> relationship given any n, then the answer is obvious. >>>>>> Do "state the relationship given any n"... I mean, what is it, exactly? >>>>>> >>>>> Uh, here it is again. in(n)=10n. out(n)=n. contains(n)=in(n)-out(n)=9n. >>>>> lim(n->oo: contains(n))=oo. Basta cosi? >>>> >>>> What is in(n)? The sets I and everyone but you are talking about are >>>> IN = { n | -1/2^(floor(n/10)) < 0 } >>>> OUT = { n | -1/2^n < 0 } >>>> Noone has ever mentioned or defined in(n) >>>> >>>> What is the definition of in(n)? Is is a set? >>>> >>>> Stephen >>>> >>> out(n) is the number of balls removed upon completion of iteration n, >>> and is equal to n. >> >>> in(n) is the number of balls inserted upon completion of iteration n, >>> and is equal to 10n. >> >> But there are no balls or iterations in the problem I posed. >> So why do you keep talking about balls and iterations? >> Are you really that incapable of participating in a discussion? >> > Are you incapable of responding to a particular formulation of the > problem? Did you start the discussion? What makes you think you can > steer it? If you did not want to answer the question I posted, you should not have responded to my post. Stephen |