An uncountable countable set
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From: Virgil on 30 Oct 2006 16:33 In article <454630cf(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Virgil wrote: > >> In article <45439743(a)news2.lightlink.com>, > >> Tony Orlow <tony(a)lightlink.com> wrote: > >> > >>> The purpose being to try to obscure details of the stated problem. > >> That does seem to be TO's purpose. > >> > >> The stated problem was [ with one modification]: > >> Given an initially empty vase. > >> Given the infinite set of finite natural numbers, staring with 1, and a > >> ball with each number marked on it. > >> At times in minutes before noon: > >> at t = 1/n balls numbered 10*(n -1) +1 to 10*n are inserted into the > >> vase and then in that same instant ball n is removed. > >> [At noon, a cube is placed in the vase.] > >> What is the state of the vase at noon ? > > > > Seems to me that the cube won't fit in the vase because the vase is full > > of balls that have infinite non-standard reals marked on them. > > > > Your vase would have to be uncountably large, and you still might need > an infinitesimal cube. ;) TO's vase may be full, but mine will be empty.
From: Virgil on 30 Oct 2006 16:36 In article <4546326d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > > <snip> > > > >> t=-1/n ^ t=0 -> -1/n=0. T v F? > > > > T. > > > > However, in a mathematical argument it is equally true that (assuming t > > is a real and n a natural): > > > > t=-1/n ^ t=0 -> -1/n > 7 > > > > Cheers - Chas > > > > That's debatable. Intuitionistic logicians reject that a false premise > implies anything. As TO is no sort of logician at al, he is not in a position to argue the point.
From: Virgil on 30 Oct 2006 16:38 In article <454632a3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4543b0b3(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> The experiment occurred in [-1,0). Talk of time outside that range is > >> irrelevant. Times before that are imaginary, and times after that are > >> infinite. Only finite times change anything, so if something changes, > >> it's at a finite, negative time. > > > > Then let us change the experiment to include the insertion into the vase > > of a cube at one minute after noon. > > > > The experiment now ranges over [-1,1]. > > > > What are the contents of the vase at times in [0,1), TO? > > > > An uncountable number of balls, all infinitely numbered. As none of them exist in the original problem, where does TO get them from? And how does he manage to make them come into existence on his command? Such magic is no part of mathematics. > > TO is confused! Still or again? probably still, as there doesn't seem to > > be much time at which he is not.
From: Virgil on 30 Oct 2006 16:41 In article <45463357(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> I am beginning to realize just how much trouble the axiom of > >>>> extensionality is causing here. That is what you're using, here, no? The > >>>> sets are "equal" because they contain the same elements. That gives no > >>>> measure of how the sets compare at any given point in their production. > >>>> Sets as sets are considered static and complete. However, when talking > >>>> about processes of adding and removing elements, the sets are not > >>>> static, but changing with each event. When speaking about what is in the > >>>> set at time t, use a function for that sum on t, assume t is continuous, > >>>> and check the limit as t->0. Then you won't run into silly paradoxes and > >>>> unicorns. > >>> There is a lot of stuff in there. Let's go one step at a time. I believe > >>> that one thing you are saying is this: > >>> > >>> |IN\OUT| = 0, but defining IN and OUT and looking at |IN\OUT| is not the > >>> correct translation of the balls and vase problem into Mathematics. > >>> > >>> Do you agree with this statement? > >> Yes. > > > > OK. Since you don't like the |IN\OUT| translation, let's see if we can > > take what you wrote, translate it into Mathematics, and get a > > translation that you like. > > > > You say, "When speaking about what is in the set at time t, use a > > function for that sum on t, assume t is continuous, and check the limit > > as t->0." > > > > Taking this one step at a time, first we have "use a function for that > > sum on t". How about we use the function V defined as follows? > > > > For n = 1,2,..., let > > > > A_n = -1/floor((n+9)/10), > > R_n = -1/n. > > > > For n = 1,2,..., define a function B_n by > > > > B_n(t) = 1 if A_n <= t < R_n, > > 0 if t < A_n or t >= R_n. > > > > Let V(t) = sum_n B_n(t). > > > > Next you say, "assume t is continuous". Not sure what you mean. Maybe > > you mean assume the function is continuous? However, it seems that > > either the function we defined (e.g., V) is continuous or it isn't, > > i.e., it should be something we deduce, not assume. Let's skip this for > > now. I don't think we actually need it. > > > > Finally, you write, "check the limit as t->0". I would interpret this as > > saying that we should evaluate the limit of V(t) as t approaches zero > > from the left, i.e., > > > > lim_{t -> 0-} V(t). > > > > Do you agree that you are saying that the number of balls in the vase at > > noon is lim_{t -> 0-} V(t)? > > > > Find limits of formulas on numbers, not limits of sets. Is there any member of IN ( = the set of balls inserted before noon) which is not a member of OUT (= the set of balls removed before noon)? Unless TO can find some member of IN which is not a member of OUT, IN\OUT = {}.
From: Virgil on 30 Oct 2006 16:43
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? I don't think so. |