An uncountable countable set
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From: David Marcus on 17 Oct 2006 20:53 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> 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? I asked if you knew the purpose of the construction of the natural numbers from sets and you reply with a question about the purpose of the von Neumann ordinals. It doesn't help if you jump from topic to topic. Should I take your reply as a "no"? -- David Marcus
From: David Marcus on 17 Oct 2006 20:58 Tony Orlow wrote: > 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. You seem to have lost all the specific numbers in your translation. Can't you include them in the English version of the problem? How would you translate the following into English? 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: David Marcus on 17 Oct 2006 21:14 Tony Orlow wrote: > Virgil wrote: > > In article <4533d315(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >>> Then let us put all the balls in at once before the first is removed and > >>> then remove them according to the original time schedule. > >> Great! You changed the problem and got a different conclusion. How > >> very....like you. > > > > Does TO claim that putting balls in earlier but taking them out as in > > the original will result in fewer balls at the end? > > If the two are separate events, sure. Not sure what you mean by "separate events". Suppose we put all the balls in at one minute before noon and take them out according to the original schedule. How many balls are in the vase at noon? -- David Marcus
From: Tony Orlow on 17 Oct 2006 21:17 Virgil wrote: > 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. According to Virgilogic. > Since the problem is given in time-wise terms and asks a time-based > question, ignoring time is ignoring the requirements of the problem. The problem is given first with a description of what happens at each iteration, and then the iterations are bijected with the moments in a Zeno machine to produce an artificial upper bound at noon, when the iterations have no LUB. If you claim they do, they you undoubtedly claim that LUB is omega, which is why I say that your interpretation of the problem rests on the notion of omega. > >> 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. There is something wrong with applying a Zeno machine to create an artificial LUB on the naturals. > > And there has been something wrong with TO's attempts at logic for a > long time. You don't understand most of what I say, or remember it for very long, so how would you know?
From: David Marcus on 17 Oct 2006 21:24
Tony Orlow wrote: > MoeBlee wrote: > > P.S. > > > > I lost the context, but somewhere you (Orlow) posted: > > > > "ZF and NBG don't handle sequences or their sums, but only unordered > > sets" > > > > Z set theory defines and proves theorems about ordered tuples, finite > > and infinite sequences, and infinite summations and infinite products > > and many other things like that. > > > > MoeBlee > > Huh! But I thought sets were unordered. If the theory of infinite series > is derived from set theory, how come they seem to contradict each other > here? The elements of a set do not have an order. However, set theory is richer than you give it credit for. For example, it is standard that one may define an ordered pair (a,b) as the set {{a},{a,b}}. Using this definition, one may prove that (a,b)=(x,y) iff a=x and b=y. It would be better to say that infinite series (and Calculus) may be based on set theory (not "derived from"). I.e., we may use set theory as our foundation and build everything we need to do Calculus (and pretty much all of Mathematics). > I don't recall a derivation or proof of the empty vase from the > axioms of set theory. That is because it is really a Calculus problem. While the steps in the proof could be traced back to the axioms in ZF, this would be a lot of work. That's why we teach students Calculus first and only work down towards the foundations later in their education. -- David Marcus |