An uncountable countable set
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From: Tony Orlow on 16 Oct 2006 21:05 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.
From: Ross A. Finlayson on 16 Oct 2006 21:19 David Marcus wrote: > MoeBlee wrote: > > David Marcus wrote: > > > Ross A. Finlayson wrote: > > > > There is no universe in ZF, ZF is inconsistent. > > > > > > What exactly is the inconsistency, please? > > > > Of course he'll never show you an inconsistency. "Set theory is > > inconsistent" is just one among the phrases that he's fond of > > muttering. > > Apparently true. > > -- > David Marcus No, I think I already have, but you won't accept it, because you're invincible, in parrradise. Also, I do not "mutter". Quantify over sets: where do they come from? If you think it's the cumulative hierarchy, the axiom of infinity says it's all of them. Quantify over sets, it's not a set. There's no universe in ZF, and there is a universe. So, ZF's implication that it describes the universe of sets is obviously wrong. Also, I argue that incompleteness is inconsistency. In pure set theory, everything's a set. Ross
From: David Marcus on 16 Oct 2006 21:25 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? -- David Marcus
From: David Marcus on 16 Oct 2006 21:29 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. -- David Marcus
From: David Marcus on 16 Oct 2006 21:32
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")? -- David Marcus |