An uncountable countable set
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From: David Marcus on 17 Oct 2006 21:26 Tony Orlow wrote: > Thanks, Stephen. I am wondering the same thing. How does one formally > prove this from the axioms of ZFC? I believe this uses AoC, and no, I > don't like it. To me, what the Axiom of Choice really should be > conveying is a dimensional aspect to set, which I think is something > it's used for, but I think it is abused beyond that. So, I have the same > question. How is this proven from the axioms? It is a straightforward Calculus problem (once you translate it into Mathematics): Problem: For n = 1,2,..., define A_n = 12 - 1 / 2^(floor((n-1)/10)), R_n = 12 - 1 / 2^(n-1). 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, undefined if t = A_n or t = R_n. Let V(t) = sum{n=1}^infty B_n(t). What is V(12)? -- David Marcus
From: Dik T. Winter on 17 Oct 2006 21:24 In article <36982$4534c6e0$82a1e228$20375(a)news1.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > MoeBlee wrote: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: .... > >>>>>Axiom of infinity: There exists a set x such that the empty set is a > >>>>>member of x and whenever y is in x, so is S(y). > >>>> > >>>>Which is actually the construction of the ordinals. Right? > >>> > >>>Wrong. > >> > >>Don't understand why that's wrong. Please explain. .... > Why then does the axiom of infinity not define the (finite) ordinals? It was not disputed that it defined the finite ordinals, it was disputed that it defined the ordinals. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Tony Orlow on 17 Oct 2006 21:32 Virgil wrote: > In article <453434b7(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> 15 is a specific finite number for which we can state its times of entry >> and exit. At its time of exit, balls 16 through 150 reside in the vase. >> For every finite n in N, upon its removal, 9n balls remain. For every n >> e N, there is a finite nonzero number of balls in the vase. Every >> iteration in the sequence is indexed with an n in N. Therefore, nowhere >> in the sequence is there anything other than a finite nonzero number of >> balls in the vase. >> >> Now, where, specifically, in the fallacy in that argument? > > The only "fallacy" is that the original question does not ask about what > happens *during* the sequence, so that none of your analysis is relevant > to the question of what the situation is after the sequence is over and > done with. > > For what the situation is *after* every step has been completed we have > to ask "Which balls are in the vase *after* every step of that sequence > of steps has been completed?" > > And to answer that question we have to ask whether there are any balls > that have not been removed in some step. > > And we must answer no. You are asking what happens *after* having counted all the elements of an infinite set. You artificially create this "after" using the Zeno paradox machine, but that is a ruse. You are claiming the unending set ends, while you accuse me of not understanding that unending sets don't end. So, you're a hypocrite. But, I think that's long since been established. >>>> Your statement concerning n does >>>> not cover noon, because noon=f(oo), and oo is outside your range. > > It may be outside your range, TO, but it is specifically within the > range of the original problem. The original problem states what happens at every naturally-indexed iteration, all of which occur before noon. It does not cover what happens AT noon, if anything, and if nothing happens at noon, then the state at noon should be the same as it was before: a non-empty vase. >>> You've lost me. >> Nothing happens at noon, if all sequential iterations are finite, given >> the time sequence. At all moments before noon, as has been conceded, >> there are a nonzero number of balls in the vase. > > And at noon every single one of them has been withdrawn. Only in Virgilogic and transfinitological philosophy. > >> Deduction depends on assumptions. Set theory's are phony in this case. > > TO is a phony in every case. I'm very genuine, and you know it. :) > > One only needs to assume what the original problem declares, a time for > insertion and a time for removal for each ball, in that order and both > before noon . The original problem states the structure of each iteration, and the time of each iteration, such that it creates an artificial LUB on the naturals. > >> At every moment before noon there are balls, and nothing happens at >> noon. Therefore, there are balls. > > Which ones? Every single ball that went in by noon has come out by noon, > so which ones are still in at noon? > Noon doesn't exist as a natural iteration of the Zeno machine, but is the combination of all infinite iterations combined. > >> I stated my argument. Refute it, stating specifically the logical error. > > TO's logical error is that ignores induction: if the first natural is in > a set and the successor of each natural is in the set then all naturals > are in the set. In this case the set is the set of numbers on the balls > removed before noon. Induction says f(0)^ (f(n)->f(s(n))) -> A n e N f(n). It says nothing about infinite values of n, which is what you have at noon. That's obvious. I recommend you stick to ZFC to prove this, specifically, from the axioms. > > So that either TO has sneaked some unnumbered balls into the vase when > no one else was looking or the vase is empty as soon as all the > removals have been completed, which is noon. > Yeah, as soon as you have reached the end of the set with no last element. >>>> Any specific finitely indexed ball has a >>>> specific finite time before noon at which it is inserted and another at >>>> which it is removed. >>> And since it is never replaced in the vase at any other time after its >>> removal, we can then conclude: "there is no ball in the vase at noon >>> which is labelled with a natural number". Go on... >> Except that with the removal of ball n, for every n in N, balls n+1 >> through 10n remain, a nonzero number. Is that not the case? > > And is not every one of them not also removed before noon? > Sure, once elements 10n+1 through 100n have been inserted. Yeah, sure, every one of those has been removed before noon too, but not until 100n+1 through 1000n have been inserted. Do you see a pattern here? >>>> The set-theoretic claim is that, even >>>> though nothing happens AT noon, nevertheless BY noon the vase is empty, >>>> even though BEFORE noon there are potentially infinitely many balls in >>>> the vase. > > There is no time at which there are infinitely many balls in the vase. > At noon, if noon exists, there is some infinite number of balls in there, 9/10 as many as the infinite number which have been inserted. > > >>> There is a misstatement in your assertion (the number of balls in the >>> vase is always finite at any t<0); but I think your main objection is >>> in fact linguistic and not mathematical. >> Think again. It has to do with the Zeno machine and the Twilight Zone >> called omega. > > No, it has to do with mathematical induction. See above. Yeah. I saw. That covers every time BEFORE noon. Try again. > >> Suppose, for a second, that if anything happens AT 0, and infinite >> number of things happen at the same moment. > > > According to the original problem, everything happens before noon. > Then noon doesn't exist in the original problem, especially if you claim it has anything to do with standard inductive proof. Of course, with my finite-case induction, it's easily provable that for all n, 9n balls remain after iteration n, and that for n>0, 9n>0. > >>> X at time t, and f(t) = 0 if nothing happens to X at time t. >> The question is whether anything happens at t=0. > > If t = 0 means noon, its all over by then. Yep
From: Tony Orlow on 17 Oct 2006 21:37 Ross A. Finlayson wrote: > Virgil wrote: >> In article <45343843(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >> >>> Virgil wrote: >>>> In article <4533d18b(a)news2.lightlink.com>, >>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> >>>>> Virgil wrote: >>>>>> In article <45319b8c(a)news2.lightlink.com>, >>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> >>>>>>> Given any finitely numbered ball, we can calculate its entry and exit >>>>>>> times. However, we can also say that when it exits, there are more balls >>>>>>> in the vase than when it entered. If you had any upper bound to your set >>>>>>> of naturals, you'd see your logic makes no sense, but there is none. >>>>>> When expressed as functions of time, rather than the number of >>>>>> operations, there is no problem with having an empty vase at noon. >>>>> When expressed in terms of iterations, the conclusion is quite the >>>>> opposite, so you cannot claim to be working with pure unquestionable >>>>> logic. You must choose which logical construction of the two is valid, >>>>> if either. >>>> As the problem is stated in terms of the times at which events occur, >>>> expressing things in terms of time is natural and indicated. >>>> >>>> When expressed in terms only of events, there is no longer any >>>> requirement that there even be an event of completing all the >>>> insertions-removals, or if there is, that it be close in time to any >>>> other event. You cannot dismiss time and still have it. >>> You can say, "What if we do this an infinite number of times?", or, "If >>> we do this n times, how many balls are in the vase?" >> In the original problem each action is specifically linked to time and >> the question asked is also specifically linked to time. >> >> So that TO's , and other's, attempts to ignore time as an essential part >> of the problem are trying to obscure the problem. >>>>>>> Before noon, there are balls. At noon, there are not. What happened? >>>>>> They were one by one removed. >>>>> "One by one" and all removed, but no last one. Vigilogic at its worst. >>>> It is apparently TOlogic, that if one moves from point A to point B one >>>> must first cover half the distance then half the remaining distance, and >>>> so on ad infinitum, so that one never reaches point B. >>>> It is Virgilogic that one can go from one such midpoint to the next "one >>>> by one" but still reach point B in finite time. >> 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
From: Tony Orlow on 17 Oct 2006 21:40
Ross A. Finlayson wrote: > Tony Orlow wrote: >> Virgil wrote: >>> In article <4533d18b(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <45319b8c(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Given any finitely numbered ball, we can calculate its entry and exit >>>>>> times. However, we can also say that when it exits, there are more balls >>>>>> in the vase than when it entered. If you had any upper bound to your set >>>>>> of naturals, you'd see your logic makes no sense, but there is none. >>>>> When expressed as functions of time, rather than the number of >>>>> operations, there is no problem with having an empty vase at noon. >>>> When expressed in terms of iterations, the conclusion is quite the >>>> opposite, so you cannot claim to be working with pure unquestionable >>>> logic. You must choose which logical construction of the two is valid, >>>> if either. >>> As the problem is stated in terms of the times at which events occur, >>> expressing things in terms of time is natural and indicated. >>> >>> When expressed in terms only of events, there is no longer any >>> requirement that there even be an event of completing all the >>> insertions-removals, or if there is, that it be close in time to any >>> other event. You cannot dismiss time and still have it. >> You can say, "What if we do this an infinite number of times?", or, "If >> we do this n times, how many balls are in the vase?" >> >>>>>> Before noon, there are balls. At noon, there are not. What happened? >>>>> They were one by one removed. >>>> "One by one" and all removed, but no last one. Vigilogic at its worst. >>> It is apparently TOlogic, that if one moves from point A to point B one >>> must first cover half the distance then half the remaining distance, and >>> so on ad infinitum, so that one never reaches point B. >>> It is Virgilogic that one can go from one such midpoint to the next "one >>> by one" but still reach point B in finite time. >> Zeno's paradox is long since explained. This one is too. > > Tony, you got something wrong: it's not "Vigilogic", it's "VBN." Where there is TOmatics, there's always some Virgilogic to annoy it. What's "VBN"? Vase-ball-neurosis? Virgil being numbskull? :) > > I think Toni understands I'm a post-Cantorian. Indubitably. :) > > Tony, I think for Virgil to call you a phony, ignore it. There are > plenty of people who'll talk to you, as we all read these discussions > until we get bored. I don't take guard dogs growls personally. > > Zeno's paradox appeases. > > Look it up in the dictionary. > > Ross > I'm aware of it, and its role in the vase-ball-neurosis. Tony |