An uncountable countable set
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From: Ross A. Finlayson on 18 Oct 2006 00:30 Tony Orlow wrote: > Ross A. Finlayson wrote: > > Virgil wrote: .... > >> It is something that I manage to do every time I move. It has never yet > >> taken me infinitely long to get to any point I have been to any other > >> point I have ever got to. > >> > >> However it does seem to be taking TO an infinitely long time to reach > >> common sense. Perhaps it is just too far for him to reach. > > > > Try taking nine steps backwards for each forwards. > > > > Virgil, people put up with you. Don't worry, I know I'm unliked. > > > > Do you know anything about physics? I just wonder if you ever heard > > the story of why one dimension of time was sufficient. > > > > Ross > > > > Tell the story, Ross. I can't even conceive of more than one time > dimension in the universe, although an infinite number of spatial > dimensions makes sense. Time is the direction of expansion is all, and > always perpendicular to the expanding space. But, what's your story? > > TOny Hi Tony, Well, yes, as a matter of fact I do have an opinion about that. Basically it seems that there's a notion of more than one dimension of time, it's basically similar in vein to a paradox of Zeno. That's about how "infinitesimals have infinitesimals". In resolution, one dimension, or direction, of time is sufficient, that they don't. I search for "dimensions of time" in posts I have written and there is a decent discussion of layman perceptions of quantum mechanics. Did you know a report of NASA says HUP doesn't hold? There was a fellow for a while being promoted as having some new ideas about time that I have not heard of since then. VBN is coffee table book nonsense physics of last millenium. The balls and vase are physical things. Accepting that situation means I have a uniform random natural number generator, and a Maxwell demon that basically vaporizes the entire experiment well before noon. Not counting measure theory, where "transfinite" cardinality is just used grossly and qualitatively where other much more specific means are better suited, as there are sufficient "non-standard" measure theories, the transfinite cardinals remain unused in physics. Ross
From: Virgil on 18 Oct 2006 00:36 In article <453589db(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45343e6f(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> David Marcus wrote: > >>>>>>>>> Tony Orlow wrote: > >>>>>>>>>> David Marcus wrote: > >>>>>>>>>>> How about this problem: Start with an empty vase. Add a ball to a > >>>>>>>>>>> vase > >>>>>>>>>>> at time 5. Remove it at time 6. How many balls are in the vase at > >>>>>>>>>>> time > >>>>>>>>>>> 10? > >>>>>>>>>>> > >>>>>>>>>>> Is this a nonsensical question? > >>>>>>>>>> Not if that's all that happens. However, that doesn't relate to > >>>>>>>>>> the > >>>>>>>>>> ruse > >>>>>>>>>> in the vase problem under discussion. So, what's your point? > >>>>>>>>> Is this a reasonable translation into Mathematics of the above > >>>>>>>>> problem? > >>>>>>>> I gave you the translation, to the last iteration of which you did > >>>>>>>> not > >>>>>>>> respond. > >>>>>>>>> "Let 1 signify that the ball is in the vase. Let 0 signify that it > >>>>>>>>> is > >>>>>>>>> not. Let A(t) signify the location of the ball at time t. The > >>>>>>>>> number > >>>>>>>>> of > >>>>>>>>> 'balls in the vase' at time t is A(t). Let > >>>>>>>>> > >>>>>>>>> A(t) = 1 if 5 < t < 6; 0 otherwise. > >>>>>>>>> > >>>>>>>>> What is A(10)?" > >>>>>>>> Think in terms on n, rather than t, and you'll slap yourself awake. > >>>>>>> Sorry, but perhaps I wasn't clear. I stated a problem above in > >>>>>>> English > >>>>>>> with one ball and you agreed it was a sensible problem. Then I asked > >>>>>>> if > >>>>>>> the translation above is a reasonable translation of the one-ball > >>>>>>> problem into Mathematics. If you gave your translation of the > >>>>>>> one-ball > >>>>>>> problem, I missed it. Regardless, my question is whether the > >>>>>>> translation > >>>>>>> above is acceptable. So, is the translation above for the one-ball > >>>>>>> problem reasonable/acceptable? > >>>>>> Yes, for that particular ball, you have described its state over time. > >>>>>> According to your rule, A(10)=0, since 10>6>5. Do go on. > >>>>> > >>>>> OK. Let's try one in reverse. First the Mathematics: > >>>>> > >>>>> > >>>>> Let B_1(t) = 1 if 5 < t < 7, > >>>>> 0 if t < 5 or t > 7, > >>>>> undefined otherwise. > >>>>> > >>>>> Let B_2(t) = 1 if 6 < t < 8, > >>>>> 0 if t < 6 or t > 8, > >>>>> undefined otherwise. > >>>>> > >>>>> Let V(t) = B_1(t) + B_2(t). What is V(9)? > >>>>> > >>>>> > >>>>> Now, how would you translate this into English ("balls", "vases", > >>>>> "time")? > >>>> That's not an infinite sequence, so it really has no bearing on the vase > >>>> problem as stated. I understand the simplistic logic with which you draw > >>>> your conclusions. Do you understand how it conflicts with other > >>>> simplistic logic? It's the difference between focusing on time vs. > >>>> iterations. Iteration-wise, it never can empty. There's something wrong > >>>> with your time-wise logic which has everything to do with the Zeno > >>>> machine and the indistinguishability of iterations outside of N. For > >>>> every n e N, iteration n results in 9n balls left over, a nonzero > >>>> number. If all steps are indexed by n in N, then this result holds for > >>>> the entire sequence. Within your experiment, with ball numbers all in N, > >>>> you never reach noon, and at every moment for the minute before it the > >>>> vase is nonempty. > >>> I didn't say it was an infinite sequence nor did I say it had a "bearing > >>> on the vase problem as stated". However, I asked you a question. I don't > >>> believe you answered my question. So, let me try again: > >>> > >>> How would you translate the mathematical problem I wrote above into > >>> English ("balls", "vases", "time")? > >>> > >> We could say we insert one ball, then another, then remove one, then the > >> other, and how many balls are in the vase after that? 0. That's the > >> sequence of events and the result. > > > > Now merely repeat similarly one insertion and one removal for each n in > > N to get the original vase problem. > > Oh, did you forget the order of events, where ten are added as each is > removed? That didn't occur in the irrelevant gedankenette that David > offered. As the only important part of that is that each nth ball, for n in N, is inserted before being removed and removed before noon. Absolutely ANY arrangement of insertions and removals satisfying those constraints will leave the vase empty at noon.
From: Virgil on 18 Oct 2006 01:06 In article <453593c4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. It is a set with one member. Which creates the potentiality of there somewhere being a set with two members. And the first thing one knows, one has counting.
From: Virgil on 18 Oct 2006 01:09 In article <4535944f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. Naw, that's 1.0, not 1, that you are talking about. Your 1.0 is only good for measuring, but no good for counting.
From: Virgil on 18 Oct 2006 01:18
In article <4535970a(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: > >>>>>>>>>> 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? Given an arbitrary set of individually identifiable balls, S, and an empty vase capable of containing then all. Assume on some day that for each ball in S there is a time before noon at which that ball is placed in the vase and a later time, also before noon at which that ball is removed from the vase. Regardless of the arrangement of the times of insertion or removal of individual balls, so long as they all satisfy those assumptions, the vase will be empty at noon of that day. > > > > > * 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> |