An uncountable countable set
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From: David Marcus on 17 Oct 2006 21:41 Han de Bruijn wrote: > MoeBlee wrote: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > >>MoeBlee schreef: > >> > >>>Han de Bruijn wrote: > >>> > >>>>Virgil 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. > > > > It's not the definition of 'ordinal' and there are ordinals that are > > not "constructed" or proven to exist by the axiom of infinity. > > Huh? > > http://www.jboden.demon.co.uk/SetTheory/ordinals.html > > What I see there is that the empty set is a member of the (finite) > ordinals, because 0 = { } . Right? > > http://en.wikipedia.org/wiki/Axiom_of_infinity > > We define the successor S(y) of y as y u { y } . Right? > > Now let { } be a member of the finite ordinals, then also { } u {{ }} > is a member of the finite ordinals, hence the set {{},{{}}, .. }. Right? > > Why then does the axiom of infinity not define the (finite) ordinals? You didn't say "finite" before. First of all, an axiom isn't a definition. Second of all, the set that the axiom of infinity says exists could contain the set of natural numbers as a proper subset. > > The axiom of infinity is just not the construction of the ordinals. > > > Why don't you just read a set theory textbook rather than remain > > ignorant about that which you are so opinionated. > > No matter how much textbooks about set theory I read, it all remains > abacedabra for me. I just looked in two books on set theory on my shelf and both explicitly state that the set in the axiom of infinity need not be the natural numbers. The books are "Naive Set Theory" by Paul R. Halmos and "Set Theory, An Introduction to Independence Proofs" by Kenneth Kunen. Which math textbooks have you read? Have you worked the problems in the ones you've read? Taken any math courses at a university? -- David Marcus
From: Tony Orlow on 17 Oct 2006 21:50 Virgil wrote: > In article <45343a40(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <4533d315(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <45319e2d(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Virgil wrote: >>>>>>> In article <453108b5(a)news2.lightlink.com>, >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>> >>>>>>>>> So how many balls are left in the vase at 1:00pm? >>>>>>>>> >>>>>>>> If you paid attention to the various subthreads, you'd know I just >>>>>>>> answered that. Where the insertions and removals are so decoupled, >>>>>>>> there >>>>>>>> is no problem. Where the removal of a ball is immediately preceded and >>>>>>>> succeeded by insertions of 10, the vase never empties. >>>>>>> It may that it never "empties", but at noon, and thereafter, it has >>>>>>> "become empty". >>>>>>> >>>>>> And when did that happen? >>>>> At the invisible transition from forenoon to noon. >>>> Oh, you mean the moment between all moments before noon and the moment >>>> of noon? >>> >>> No! The set of all moments before noon is, in the real number model of >>> times, an open set whose least upper bound, noon, is not a member of >>> that set but such that there are no times between that set and its LUB, >>> noon. >>> So that as sets of times, the forenoon and what comes after, form a >>> partition of times into disjoint sets with every time being in one or the >>> other but none in both. >>> >> So, something happens at this partition, which is not actually a point >> in time, but a conceptual separation between this point and everything >> that came before it? > > One can separate the reals into everything before 0 as one set and 0 > and everything after it as the other. Does TO claim time is less > seperable? > Linear time? no. > >> Is this your quantum math again? Do we step out of >> time for less than a moment to make the balls disappear? > > Any partition of time into before and after must have a point of > separation, like the noon of this case. Everything happens before that > point. Yeah, noon doesn't exist in the description of the problem. It's like saying, "Everyone on Earth has 3 children which survive, for four generations, and then half the population of the planet dies. This happens an infinite number of times. What happens when there is no more planet?" The question is itself a non-sequitur. >>>>>>> The only necessary constraint on insertions of balls into the vase and >>>>>>> removals of balls from the vase is that each ball that is to be removed >>>>>>> must be inserted before it can be removed, and, subject only to that >>>>>>> constraint, the set of balls remaining in the vase at the end of all >>>>>>> removals is independent of both the times of insertion and of the times >>>>>>> of removal. >>>>>>> >>>>>>> To argue otherwise is to misrepresent the problem. >>>>>> You already said that.....WRONG!!!! >>>>> What is wrong about it? >>>> This: >>>> >>>>>> There is the additional constraint that, before removing any ball, ten >>>>>> have been inserted. >>>>> >>>>> 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. > > Now that is really illogical: TO claims that having the balls in for > longer times will leave fewer of them at the end. I claim that having balls stop being inserted sooner leads to fewewr balls left, yes. >>> If so, by what logic, and if not, what difference does it make? >> That the removal of balls in the second phase is unrelated to the >> addition of balls in the first stage. In the original problem, no ball >> is removed with the immediately preceding addition of ten. > > >>>>> Does TO claim that by putting balls in earlier there can be at ANY time >>>>> fewer balls in the vase that when putting them in later? >>>> Yes, I've already explained that. > > Not to the satisfaction of anyone but yourself. Speak for yourself. >>> Not to anyone else's satisfaction. >> I'm sure to that of some. You're never satisfied except with yourself. > > I am not satisfied that having more balls in for longer times gives less > balls in at any time. How about NOT adding balls any more after a given time, and only removing them after that? That's why the vase empties in that case. > > One certainly starts with more balls. At what time do more balls become > less balls? And why? When the net addition of nine balls overtakes the mere subtraction of one. >>>>> But when one puts them all in early enough, it becomes obvious that the >>>>> vase must be empty at noon. >>>> Yes, if all insertions occur before all removals. >>> How does that change things? Is there any time at which putting balls in >>> earlier forces fewer balls to be in the vase? If so, at what time does >>> the number of balls from the earlier insertions become less that the >>> number of balls from the later insertions. >>> >>> As far as I can see, putting balls in earlier can only increase the >>> number of balls in the vase at some times, but not decrease the number >>> at any time. >>> >>> >> Why, did you put more balls in, when you added them earlier? > > When some balls are to be added earlier then at all times there will be > at least as many balls as there would be when entering them later. > Anything else is idiotic. When ALL balls are added, and then balls are ONLY removed, to say that gives the same result as repeatedly adding more balls than you remove, that's what's idiotic, to borrow your obnoxious term. > >> You really >> don't understand the implications of the Zeno machine, do you? > > I do not understand how having more balls in the vase for longer times > can produce less balls in the urn at any time. There is so much you fail to understand, or succeed in misunderstanding, that I don't even know where to begin with you. If you can't grasp the logic here, I really do
From: Tony Orlow on 17 Oct 2006 21:53 Virgil wrote: > In article <45343aed(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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. > > TO's versions of set theory all required ordered sets, so why is he now > objecting to ordered sets? I'm throwing your repeated objections back atcha. > > And certainly N is ordered and well ordered, as are infinite sequences, > series and products. > Then why object to my use of natural quantitative order when comparing infinite sets of reals formulaically? Again, you're just a hypocrite who spews whatever argument happens to be handy at the time. > >> If the theory of infinite series >> is derived from set theory, how come they seem to contradict each other >> here? > > They do not seem to for anyone who understands them. For incompetents > like TO, all sorts of perfectly natural and logical things may seem to > be what they are not. > Again with the insults. >> I don't recall a derivation or proof of the empty vase from the >> axioms of set theory. > > TO has a lot of practice at forgetting important derivations and proofs. Which axioms are involved? You don't even know. Upon which axioms of ZFC is the measure of cardinality based?
From: Tony Orlow on 17 Oct 2006 21:56 Virgil wrote: > In article <45343e6f(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: >>>>>>>>> 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. > > Now merely repeat similarly one insertion and one removal for each n in > N to get the original vase problem. Oh, did you forget the order of events, where ten are added as each is removed? That didn't occur in the irrelevant gedankenette that David offered.
From: David R Tribble on 17 Oct 2006 22:01
Tony Orlow wrote: >> The sequence of events consists of adding 10 and removing 1, an infinite >> number of times. In other words, it's an infinite series of (+10-1). > David Marcus wrote: >> Sorry, but I don't quite understand. When you stated the problem in >> English, it ended with a question mark. But, your statement in >> Mathematics does not end with a question mark. If it is a >> problem/question, I think it should end with a question mark. Please >> give the statement of the problem in Mathematics. > Tony Orlow wrote: > What is sum(n=1->oo: 9)? Because 9 = 10 - 1, it obviously must be true that sum{n=1 to oo} 9 is equal to [sum{n=1 to oo} 10] - [sum{n=1 to oo} 1] Did I get it right, Tony? |