An uncountable countable set
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From: Tony Orlow on 17 Oct 2006 22:54 David Marcus wrote: > 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? > empty.
From: Virgil on 17 Oct 2006 23:41 In article <453580b1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. Compare that to TO-illogic. Virgilogic conforms in all respects to standard logic and standard mathematics and TO-illogic conflicts with both. So whom besides TO himself will opt for TO-illogic ? > > > 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 each iteration is tied to a specific time. > and then the iterations are bijected with the moments in a > Zeno machine Sane people call such machines "clocks". > 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, It is the times which have a LUB, and that LUB is noon. > which is why I say that your interpretation of the > problem rests on the notion of omega. Does TO claim that every noon is "omega"? > > > > >> There's something wrong > >> with your time-wise logic Since the original problem links all events to the times at which they are to occur, and asks a question based on time, there is something wrong with any analysis, like TO's, that eliminates all references to time in trying to answer that question. > > 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? As virtually each of TO's claims contains sufficient internal evidence to demonstrate its errors, long memory is not required.
From: Virgil on 18 Oct 2006 00:06 In article <4535843c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. Actually, it is the problem itself which creates the "after", as it specifically asks about the situation at a time after all the insertions and removals have been completed. So TO is dreaming again about a problem that does not exist except in his imagination. > 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. There are ordered sets which have no maximal element even though they have upper bounds. The reals contain many such sets, as do the rationals. > >> 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. :) A genuine kook. > > > > > 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. Which is a perfectly natural LUB on the times. > > > > >> 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. Noon exists as the LUB of the times of transitions of the balls. > > > > >> 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 phony. Can TO deny that the set of numbers of balls removed before noon is, by valid induction, N? > That's > obvious. I recommend you stick to ZFC to prove this, specifically, from > the axioms. Induction is a part of ZFC. > > > > > 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. The set of times, (noon - 1/2^n minutes), has no last element but does have a LUB. > > > > 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? I see a pattern that declares that each ball is removed before noon. > > > > 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. Name one. > > > > No, it has to do with mathematical induction. See above. > > Yeah. I saw. That covers every time BEFORE noon. Try again. The set of numbers on balls nserted before noon is N The set of numbers on balls removed after insertion but before noon is also N. So that N\N = {}. > > > > >> 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. I see nothing in the original problem that would stop time.
From: Virgil on 18 Oct 2006 00:20 In article <4535884c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > 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. Does TO claim that anything in the statement of the problem justifies an assumption of non-linear time. > > 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. One can claim that the vase problem cannot occur in any physical sense, but if one accepts the statement of the problem, the only conclusion which does not require assumptions not inherent in the problem itself is that at noon the vase is empty. > > 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. Non sequitur. > > 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. When one starts with all infinitely many balls in the vase and then balls are removed on the original schedule, there will be infinitely many in the vase at all times from that group insertion up to but not including noon. > > > > >> You really > >> don't understand the implications of the Zeno machine, do you? As I am not using one, that is irrelevant. > > > > 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 don't see any hope for you. I can grasp logic well enough, but from TO I have not seen any.
From: Virgil on 18 Oct 2006 00:27
In article <45358929(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > 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? Until I see a complete axiom system based on TO's notions of quantitative order and formulaic comparison, I will stick to ZFC based reals. > Again, you're just a hypocrite who > spews whatever argument happens to be handy at the time. My arguments are all based on the statement of the problem and the consequences of ZFC and forma logic. Which cannot be said of TO's arguments. > > > > >> 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. When TO insults me, as he has done by, e.g., calling me a hypocrite above, I feel free to return the favor. > > >> 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? Cardinality, beyond the cardinality of the empty set, is not required for the vase problem. |