An uncountable countable set
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From: Tony Orlow on 31 Oct 2006 10:35 David R Tribble wrote: > Virgil wrote: >>> Are the properties of "Finlayson Numbers" known to anyone except >>> Ross himself? > > Tony Orlow wrote: >> Uh, yeah, I think I understand what his numbers are. Perhaps you've seen >> our recent exchange on the matter? They are discrete infinitesimals such >> that the sequence of them within the unit interval maps to the naturals >> or integers on the real line. Is that about right, Ross? > > Only a countable infinity of them? Then the number of infinitesimals > in [0,1] is exactly the same as the reciprocals 1/n for every natural > n>0, right? But there are c reals in [0,1], so are there more reals > than infinitesimals? > I think Ross has to answer that one. In my book, the naturals are really *N, the hypernaturals, and so there are an uncountably, actually infinite, number of them, and then EF works for me as a special case of IFR. Tony
From: Tony Orlow on 31 Oct 2006 10:46 Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> Randy Poe wrote: >>>>> Tony Orlow wrote: >>>>>> Virgil wrote: >>>>>>> In article <4542201a(a)news2.lightlink.com>, >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>> >>>>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>>>> When you say "noon doesn't occur"; I think "he doesn't accept (1): by a >>>>>>>>> time t, we mean a real number t" >>>>>>>> That doesn't mean t has to be able to assume ALL real numbers. The times >>>>>>>> in [-1,0) are all real numbers. >>>>>>> By what mechanism does TO propose to stop time? >>>>>> By the mechanism of unfinishablility. >>>>> But that's why I asked you a question about variables labelling >>>>> times yesterday, when noon clearly occurred. >>>> 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. >>>> >>>>> I can define a list of times t_n = noon yesterday - 1/n seconds, >>>>> for all n=1, 2, 3, ... >>>> Are there balls in the vase for t<-1? No. >>> What balls? What vase? >>> >>> I'm naming times. They're just numbers. >>> >>>>> Clearly this list of times has no end. But didn't noon happen? >>>> Nothing happened at noon to empty the vase, \ >>> What vase? Why are you obsessed with vases? >>> >>> Do you deny me the ability to create a set of variables >>> t_n, n = 1, 2, ...? Why do vases have to come into it? >> I thought we were trying to formulate the problem. > > No, we (some of us) are trying to formulate a completely > different problem, with balls and vases (possibly even > times) explicitly removed so that other aspects can be examined. > > Yet you keep trying to put balls and vases back in, after being told > that they are not present in the new problem. I am pointing out that your formulation doesn't match the original problem. You say above you are possibly eliminating times form the problem. Sorry, but that's a required feature, and if your mathematical description ignores them, then it's missing the boat. > > I'm asking a question not having to do with balls and vases > but that does involve times. I'm trying to get at this "noon doesn't > happen" concept and what about the problem parameters > makes you think the experiment can stop time. In another thread, > even time has been removed and the discussion is simply about > subsets of the natural numbers. That approach does not represent the problem. Set theory without times cannot describe this unending process. When you include time, this becomes clear. > > In both cases, the idea is to get AWAY from the balls and > vases problem and focus on specific, separate properties of > that problem. Without balls or vases. OK? > > - Randy > As long as it's true to the problem. I am not interested in whether you can concoct some OTHER experiment where I'll agree the vase is empty. That's a waste of time. The problem is with the original problem, and changing the problem to suit your answer is non-logical.
From: Tony Orlow on 31 Oct 2006 10:49 Virgil wrote: > 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. Coming from you, that's rich!
From: Randy Poe on 31 Oct 2006 10:49 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Randy Poe wrote: > >>>>> Tony Orlow wrote: > >>>>>> Virgil wrote: > >>>>>>> In article <4542201a(a)news2.lightlink.com>, > >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>> > >>>>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>>>>>> When you say "noon doesn't occur"; I think "he doesn't accept (1): by a > >>>>>>>>> time t, we mean a real number t" > >>>>>>>> That doesn't mean t has to be able to assume ALL real numbers. The times > >>>>>>>> in [-1,0) are all real numbers. > >>>>>>> By what mechanism does TO propose to stop time? > >>>>>> By the mechanism of unfinishablility. > >>>>> But that's why I asked you a question about variables labelling > >>>>> times yesterday, when noon clearly occurred. > >>>> 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. > >>>> > >>>>> I can define a list of times t_n = noon yesterday - 1/n seconds, > >>>>> for all n=1, 2, 3, ... > >>>> Are there balls in the vase for t<-1? No. > >>> What balls? What vase? > >>> > >>> I'm naming times. They're just numbers. > >>> > >>>>> Clearly this list of times has no end. But didn't noon happen? > >>>> Nothing happened at noon to empty the vase, \ > >>> What vase? Why are you obsessed with vases? > >>> > >>> Do you deny me the ability to create a set of variables > >>> t_n, n = 1, 2, ...? Why do vases have to come into it? > >> I thought we were trying to formulate the problem. > > > > No, we (some of us) are trying to formulate a completely > > different problem, with balls and vases (possibly even > > times) explicitly removed so that other aspects can be examined. > > > > Yet you keep trying to put balls and vases back in, after being told > > that they are not present in the new problem. > > I am pointing out that your formulation doesn't match the original > problem. It's a new problem. Are you capable of contemplating a second problem, throwing away balls and vases and starting from scratch, asking different questions about a different problem? - Randy
From: Tony Orlow on 31 Oct 2006 10:52
David Marcus wrote: > Tony Orlow 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. > > I have no clue what you mean. There are no "limits of sets" in what I > wrote. > >> Here's what I said to 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. >> >> contains(n) is the number of balls in the vase upon completion of >> iteration n, and is equal to in(n)-out(n)=9n. >> >> n(t) is the number of iterations completed at time t, equal to floor(-1/t). >> >> contains(t) is the number of balls in the vase at time t, and is equal >> to contains(n(t))=contains(floor(-1/t))=9*floor(-1/t). >> >> Lim(t->-0: 9*floor(-1/t)))=oo. The sum diverges in the limit. > > You seem to be agreeing with what I wrote, i.e., that you say that the > number of balls in the vase at noon is lim_{t -> 0-} V(t). Care to > confirm this? > No that's a bad formulation. I gave you the correct formulation, which states the number of balls in the vase as a function of t. |