An uncountable countable set
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From: Virgil on 22 Oct 2006 15:12 In article <453b3566(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> 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. > >>> I thought you just said that the vase doesn't exist at noon. If the vase > >>> doesn't exist, how can the balls be in it? > >>> > >> Either the experiment goes until noon or it doesn't. If all moments are > >> naturally indexed, then it doesn't. > > > > A set of moments is indexed. Those do not constitute > > "all moments". The clock ticks between the first and second > > insertion, for example, and we do not assign any index to > > any moment in that interval. > > > > - Randy > > > > So? Are you suggesting that something happens between all naturals > having been processed, and noon? Is there such a time between x and less > than x? All balls are processed at finite times BEFORE noon, at which > times there are a growing but finite number of balls in the vase. What > happens AT noon? Nothing. Because by then every ball has already been removed. As TO's inability to name any ball remaining at noon evidences.
From: Virgil on 22 Oct 2006 15:18 In article <453b361f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45390fcc(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> Either the experiment goes until noon or it doesn't. If all moments are > >> naturally indexed, then it doesn't. > > > > So that natural indexing won't work. > > The balls are NUMBERED WITH NATURALS! Are you daft? A good deal less daft than TO, at all events. > > > But time indexing with > > discontinuous values at each transition and at noon does. > > Yeah, time discontinuities. No, number of balls discontinuities. Time, as always, remains continuous. It is the number of balls in the vase that jumps discontinuously from one natural number value to another while time continues continuously on its merry way.
From: Virgil on 22 Oct 2006 15:21 In article <453b36c5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <453912c1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> At all times >=-1 there will be 1 ball in the vase. > > > > Which ball will that be at noon, TO? > > The last one inserted. You know, when the infinite set is finished, like > you wanted. Duh. According to the Peano, or ZF, or NBG , rules for naturals, there cannot be a last one, so that TO must be talking about some entirely different Gedanken than the one we are discussing.
From: Tony Orlow on 22 Oct 2006 15:34 Virgil wrote: > In article <453b326d(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <4539000e(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> Claimed but not justified. TO's usual technique! >> You didn't justify yours. It's clearly nonsensical. It pretends there's >> a time between noon and all times before noon. > > I only claim there is a time between any time before noon and noon. > When does the vase become empty? >>>>>>>> You really >>>>>>>> don't understand the implications of the Zeno machine, do you? >>>>> As I am not using one, that is irrelevant. >>>> So, now you're doing it in linear time? Let me know when you're done.... >>> It is the problem that uses linear time, 60 seconds to 1 minute, and so >>> on. >> The iterations do not occur in linear time. > > They occur in linear time but not at equally spaced intervals in that > time. Time being linear merely means that all times can be lined up in > order. > Yeah, and a linear function just means it looks like some kind of a line. Sure, Virgil. >>>>> I can grasp logic well enough, but from TO I have not seen any. >>>> You have to step out of the cave to see it in the light, Virgil. They're >>>> only birds...... >>> TO has obviously been standing under those birds at the wrong time too >>> often. >> heh heh. good one, Virge. > > Try showering, TO. if you use enough soap. it will even get the bird do > off. Oy.
From: Virgil on 22 Oct 2006 16:16
In article <453b44a4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> 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: > >>>>>>>>> Tony Orlow wrote: > > > > <snip> > > > >>>> You have agreed with everything so far. At every point before noon balls > >>>> remain. > >>> To be precise, the assertions above all imply that at every time t = > >>> -1/n, where n is a natural number, there are balls in the vase. > >>> > >>> But that *alone* does not even include every time t before noon; let > >>> alone every time t. For example, notice that nowhere above do you or I > >>> /explicitly/ assert: "at t=-2/3, the number of balls in the vase is a > >>> positive finite number". > >>> > >>> We assert something specific about t = -2/4, and something specific > >>> about t = -1/3, but nowhere do we directly state somthing about t = > >>> -2/3. > >>> > >>> On the other hand, given the problem statement, I think we would both > >>> /agree/ that there "should be" an obvious (perhaps even unique) > >>> well-defined answer to the question : "what is the number of balls in > >>> the vase at time t = -1/pi?" > >>> > >>> Assuming in the remaining statements that one agrees with the previous > >>> statement, this leads us to the question: what are the unstated > >>> assumptons that allow to agree that this must be the case? > >>> > >>> I attempted to describe those assumptions in my previoius post. Did you > >>> read those assumptions? If so, do you agree with those assumptions? > >> At this point I don't recall your previous post. I've been off a bit. > > > > Well, allow me to repeat them here (with two minor changes): > > > > In order to interpret the problem > > > > "At each time t = -1/n where n is a (strictly positive) natural number, > > we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the > > vase, and remove the ball labelled n from the vase. What is the number > > of balls in the vase at time t=0?" > > > > I make the following simple (and I would claim, fairly uncontroversial > > and natural) assumptions: > > > > --- (object permanence) > > > > (1) When we speak of a time t, we mean some real number t. > > > > (2) If a ball is in the vase at any time t0, there is a time t <= t0 > > for which we can say "that ball was placed in the vase at time t". > > > > (3) If a ball is placed in the vase at time t1 and it is not removed > > from the vase at some time t where t1 <= t <= t2, then that ball is in > > the vase at time t2. > > > > (4) If a ball is removed from the vase at time t1, and there is no time > > t such that t1 < t <= t2 when that ball is placed in the vase, then > > that ball is not in the vase at time t2. > > > > ---- (obedience to the problem constraints) > > > > (5) If a ball is placed in the vase at some time t, it must be in > > accordance with the description given in the problem: it must be a ball > > with a natural number n on it, and the time t at which it is placed in > > the vase must be -1/floor(n/10). > > > > (6) If a ball is removed from the vase at some time t, it must be in > > accordance with the description given in the problem: it must be a ball > > with a natural number n on it, and the time t at which it is removed > > from the vase must be -1/n. > > > > (7) If n is a natural number with n > 0, then the ball labelled n is > > placed in the vase at some time t1; and it is removed from the vase at > > some time t2. > > > > --- (very general definition of "the vase is empty at noon") > > > > (8) the number of balls in the vase at time t=0 is 0 if, and only if, > > the statement "there is a ball in the vase at time t=0" is false. > > > > --- > > > > Perhaps you would add other assumptions (9), (10), etc.; but my > > question is: > > > > Given the problem statement, do you agree that /each/ of these > > assumptions, /on its own/, is reasonable and not just some arbitrary > > statement plucked out of thin air? > > > > If not, which assumption(s) is(are) not reasonable or is(are) > > unneccessarily arbitrary? > > > > <snip> > > Those all look reasonable to me as I read them. I don't see any > statement regarding the fact that ten balls are added for every one > removed, though that can be surmised from the insertion and removal > schedule. That's the salient fact here. You never remove as many as you > add, so you can't end up empty. On the other hand, the vase can end up empty if every ball inserted into it is also removed from it. > > > > >>>> You claim nothing changes at noon. > >>> Where, exactly, above do I claim that "nothing changes at noon"? > >> Do you disagree with the other standard-bearers, and claim that > >> something DOES occur at noon? > > > > That is not a response to /my/ question "where, exactly, above do I > > claim that 'nothing changes at noon'?" > > > > /I/ don't claim that "something occurs at noon"; nor do /I/ claim that > > "nothing occurs at noon". > > Uh, what would be your opinion on the matter. CAN something occur at > noon in this experiment or not? Either way, you have a problem. Since cbrown does not express any opinion, only TO has that problem. > > > > > /YOU/ are claiming that the truth of these statements follows logically > > from our assumptions; but until you make some kind of mathematical > > statement which corresponds to "something occurs at noon", I really > > can't address it as a /mathematical/ question. > > 1/n=0. Happy? Que pasa aqui? Is n in N? How does that correspond to "something occurs at noon"? > > > > > <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? > >> I said that any specific ball was obviously out of the vase at noon. > >> > > > > That's good: we at least agree that it logically follows from (1) - (8) > > that there are no labelled balls in the vase at t=0. > > No finite balls. And in the total absence of any justification for their existence, where would any others come from? > > > > > What I honestly find baffling is |