An uncountable countable set
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From: David Marcus on 25 Oct 2006 19:04 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> So, David, you think the fact that balls leave the vase only by being > >> removed one at a time, and the fact that at all times before noon there > >> are balls in the vase, and the fact that at noon there are no balls in > >> the vase, is consistent with the fact that no balls are removed at noon? > > > >> How can you not see the logical inconsistency of an infinitude of balls > >> disappearing, not just in a moment, but at no possible moment? Are you > >> so steeped in set theory that you cannot see that an unending sequence > >> of +10-1 amounts to an unending series of +9's which diverges? What is > >> illogical about that? > > > >> In your set-theoretic interpretation of the experiment there is a > >> problem which makes your conclusion incompatible with conclusions drawn > >> from infinite series, and other basic logical approaches. > > > > I gave a Freshman Calculus interpretation/translation of the problem (no > > set theory required). Here is a suitable version: > > > > For n = 1,2,..., define > > > > A_n = -1/floor((n+9)/10), > > R_n = -1/n. > > > > 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. > > > > Let V(t) = sum{n=1}^infty B_n(t). What is V(0)? > > > > I suppose you either disagree with this interpretation/translation or > > you disagree that for this interpretatin V(0) = 0. Which is it? > > t=0 is precluded by n e N and t(n) = -1/n. Sorry, I don't follow. Were you answering my question? I gave you a choice: 1. Disagree with the interpretation/translation 2. Agree with the interpretation/translation, but disagree that V(0) = 0 Are you picking #1 or #2? > > Given my interpretation/translation of the problem into Mathematics (see > > above) and given that the "moment the vase becomes empty" means the > > first time t >= -1 that V(t) is zero, then it follows that the "vase > > becomes empty" at t = 0 (i.e., noon). > > Yes, now, when nothing occurs at noon, and no balls are removed, what > else causes the vase to become empty? No balls are added or removed at noon, but the vase becomes empty at noon. If you consider the vase becoming empty to be "something" rather than "nothing", then it is not true that nothing occurs at noon. If by "nothing occurs at noon", you mean no balls are added or removed, then it is true that nohting occurs at noon. The cause of the vase becoming empty at noon is that all balls are removed before noon, but at all times between one minute before noon and noon, there are balls in the vase. Let me ask you the same question regarding the following problem. Problem: For n = 1,2,..., let A_n = -1/floor((n+9)/10), R_n = -1/n. 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. Let V(t) = sum_n B_n(t). What is V(0)? Answer: V(0) = 0. Considering that for all n we have A_n <> 0 and B_n <> 0 and that V(t) is approaching infinity as t approaches zero from the left, what causes V(0) to be zero? -- David Marcus
From: David Marcus on 25 Oct 2006 19:06 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >> < endless reiterations of the following > > >>> The only question is "According to the rules set up in the problem, is > >>> each ball which is inserted into the vase before noon also removed from > >>> the vase before noon?" > >>> > >>> An affirmative answer confirms that the vase is empty at noon. > >>> A negative answer violates the conditions of the problem. > >>> > >>> Which answer does TO choose? > >> God, are you a broken record, or what? Let's take this very slowly. Ready? > >> > >> Each ball inserted before noon is removed before noon, but at each time > >> before noon when a ball is removed, 10 balls have been added, and 9/10 > >> of the balls inserted remain. Therefore, at no time before noon is the > >> vase empty. Agreed? > >> > >> Events including insertions and removals only occur at times t of the > >> form t=-1/n, where n e N. Where noon means t=0, there is no t such that > >> -1/n=0. Therefore, no insertions or removals can occur at noon. Agreed? > >> > >> Balls can only leave the vase by removal, each of which must occur at > >> some t=-1/n. The vase can only become empty if balls leave. Therefore > >> the vase cannot become empty at noon. Agreed? > > > > Not so fast. What do "become empty" or "become empty at" mean? > > "Not so fast"???? We've been laboring this point endlessly. The vase > goes from a state of balledness to a state of balllessness starting at > time 0. Agreed. > Balls have to have been removed for this transition to occur. Yes, but they don't have to have been removed at time 0. > >> It is not empty, and it does not become empty, then it is still not > >> empty. Agreed? > >> > >> When you bring t=0 into the experiment, if anything DOES occur at that > >> moment, then the index n of any ball removed at that point must satisfy > >> t=-1/n=0, which means that n must be infinite. So, if noon comes, you > >> will have balls, but not finitely numbered balls. In this experiment, > >> however, t=0 is excluded by the fact that n e N, so noon is implicitly > >> impossible to begin with. -- David Marcus
From: MoeBlee on 25 Oct 2006 20:17 Lester Zick wrote: > At least to > me the "well" in "well ordered" is redundant. If a set is ordered it > is "well ordered" and if not it's not ordered. Of course as you note I > haven't studied the arcana of conceptual techniques in standard set > analysis so I may be mistaken but that's how it appears. You are mistaken. And it is not an arcane point. Look, such terms as 'well ordered' are specific in mathematics and are not proposed to serve in the same role as such everyday notions as 'ordered'. And it's not that you haven't studied arcane aspects of set theory. You haven't studied ANY set theory or mathematical logic at all. A well ordering of a set is a linear ordering of the set such that each nonempty subset of the set has a least member. For example, the standard ordering of the real numbers is a linear ordering, but is not a well ordering. (And, by the way, 'linear' is not redudant either, since there are orderings that are not linear.) > The formality > and precision of definition are irrelevant when we have no way to say > whether they're true in mechanically exhaustive terms There is no consistent theory that includes expression of arithmetic of the natural numbers that can provide for an algorithm to determine the truth of sentences in the language of the theory. It would be nice if this weren't the case, but, alas, it is. Meanwhile, at least we can provide theories of which we DO have algorithms to determine whether something is or is not a proof from the axioms of the theory. > I gave you the only reference I had to my own "Epistemology 201: The > Science of Science" which so far as I know is the only demonstrable > framework for the exhaustion of truth ever achieved. Wonderful. But there's no mathematics offered in that post. MoeBlee
From: MoeBlee on 25 Oct 2006 20:20 MoeBlee wrote: > Ross A. Finlayson wrote: > > Eh, ZF is consistent with itself. That's because anything can be > > proven in an inconsistent theory, except where it would conflict with A > > theory of course, because there are only and all true statements in A > > theory. > > What benefit, what satisfactions do you derive from typing nonsense > such as above? > > (The only correct thing you said above is that an inconsistent theory > proves every statement (that is, every statement in the language of the > theory)). Correction: As to "ZF is consistent with itself", I do not mean to claim that ZF is inconsistent. MoeBlee
From: MoeBlee on 25 Oct 2006 20:32
Lester Zick wrote: > Who cares what the "method of models" defines as > true? You don't have to care. But the fact that there is such a method refutes your claim that mathematicans define truth only in terms of consistency. > What misstatements of set theory? What mischaracterizations of set > theory? I've been correcting you on certain of your misstatements and mischaracterizations. If you want those posts by me to be referenced, then have your secretary do the appropriate searches. > So how can > I or anyone misrepresent something that isn't true to begin with? Even if you claim that the theorems of set theory are not true, and even if the theorems of set theory are indeed not true, you misrepresent when you say untrue things about those theorems. If someone says "There are lions and tigers in my laundry room", then that is a false statement, but then it is also a false statement if I say that the person said, "There are giant salamanders in MoeBlee's laundry room". If you're going to claim that certain statements of set theory are not true, then you should at least be correct in describing what those statements are. > So call me on it. Just don't try to demand I acknowledge the virtue of > your views on mathematics and set analysis as revealed truth. I never demanded any such thing. MoeBlee |