An uncountable countable set
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From: Tony Orlow on 16 Oct 2006 22:19 David Marcus wrote: > Tony Orlow wrote: >> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>> Tony Orlow schreef: >>> >>>> Han de Bruijn wrote: >>>>> So the axiom of infinity says that you can get everything from nothing. >>>>> This is contradictory to all laws of physics, where it is said that you >>>>> pay a price for everything. E.g. mass and energy are conserved. >>>> Han, you can't really be looking for conservation of energy or momentum >>>> or mass in abstract mathematics, can you? This axiom basically defines >>>> the infinite linear inductive set. Given this method of generation, >>>> there should be things we can say about the set, no? >>> So to speak, Tony. In physics and economics, you can't get something >>> for nothing. Nothing just gives nothing. You must have _something_ to >>> start with. I find the idea absurd that natural numbers can be built >>> by putting curly braces around the empty set. >>> >>> Han de Bruijn >>> >> 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. > > This is not relevant since no one ever said that the set containing 0 is > an "accurate model of what 1 is". Before criticizing something, it helps > to know its purpose. So, I'll ask you: Do you know the purpose of the > construction of the natural numbers from sets? > Why don't you explain it in your own words. What is the "purpose" of the von Neumann ordinals?
From: Tony Orlow on 16 Oct 2006 22:19 David Marcus wrote: > stephen(a)nomail.com wrote: >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>> MoeBlee wrote: >>>> Tony Orlow wrote: >>>>> No, set theory confuses the issue with its concentration on omega. >>>> Oh boy, here we go again with "Set theory confuses...". Please just way >>>> which axioms of set theory you reject and which ones you use instead. >>> Good suggestion. But, what do you think the chance is that Tony will >>> actually do that? >> Tony has said that he does not like the axiom of infinity and >> the axiom of choice. Unlike many people who object to the >> axiom of infinity, Tony believes in infinite sets. He has more >> infinities than most. As for the axiom of choice, does >> the analysis of the balls in the vase problem use the axiom >> of choice? > > My statement of the problem is > > 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)? > > Determining the value of V(12) certainly doesn't need the axiom of > choice. > What axioms ARE you using, specifically?
From: Virgil on 16 Oct 2006 22:20 In article <45342c6b(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: > >>>>>>> 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. Time-wise it has to be empty at noon. Since the problem is given in time-wise terms and asks a time-based question, ignoring time is ignoring the requirements of the problem. > There's something wrong > with your time-wise logic There is something wrong with every anti-timewise logic which tries to morph a time-wise problem into a time-free problem. And there has been something wrong with TO's attempts at logic for a long time.
From: Tony Orlow on 16 Oct 2006 22:22 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.
From: Virgil on 16 Oct 2006 22:47
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. > > >> 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. > > > > 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. > Deduction depends on assumptions. Set theory's are phony in this case. TO is a phony in every case. 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 . > 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? > > 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. 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. > > > >> 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? > >> 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. > > > > 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. > > 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. > > 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. |