An uncountable countable set
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From: Virgil on 27 Oct 2006 00:25 In article <45417d3e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> t=0 is precluded by n e N and t(n) = -1/n. > >>> Really? > >>> > >>> I hope you will accept as true that noon occurred yesterday. > >>> > >>> Let's define noon yesterday as t=0. Now let's define a set of values > >>> t_n = -1/n seconds for n=1, 2, 3, ... , that is, for all FINITE > >>> natural numbers n. > >>> > >>> Has my giving these names to those times somehow > >>> precluded noon yesterday from occurring? Retroactively? > >>> > >> Do you live in the gedanken? Oy. Nothing happens at noon. > > > > Did noon occur? > > Not within the constraints of the experiment. Nothing is allowed to > happen at noon. Then nothing is allowed to happen at all, since until noon can happen nothing in time measured from noon can happen. > > > > >> Your desired result does not happen before noon. > > > > What desired result? I didn't have an experiment, I > > just named a bunch of times. Is noon "precluded" by > > my defining that countable set of variables? > > > > - Randy > > > > Can anything happen at noon? What can change, in the vase, at noon? What > is the state before noon? If noon happens then the vase is empty at noon. If noon doesn't happen, nothing timed by a non-event occurs either.
From: imaginatorium on 27 Oct 2006 00:27 MoeBlee wrote: > Lester Zick wrote > > even more sheer brilliance! <...> > I won't quote more that opening and close, as one can become > overwhelmed by so much wisdom from just one man in just one day. Ah, I see you've noticed. Meanwhile, "Have you tried searching the archive for Zick + transcendental?" http://groups.google.com/group/sci.math/search?group=sci.math&q=zick+transcendental&qt_g=1&searchnow=Search+this+group Brian Chandler http://imaginatorium.org
From: cbrown on 27 Oct 2006 00:28 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > > > > <snip> > > > >>> My question : what do you think is in the vase at noon? > >>> > >> A countable infinity of balls. > >> > >> This is very simple. Everything that occurs is either an addition of ten > >> balls or a removal of 1, and occurs a finite amount of time before noon. > >> At the time of each event, balls remain. At noon, no balls are inserted > >> or removed. > > > > No one disagrees with the above statements. > > > >> The vase can only become empty through the removal of balls, > > > > Note that this is not identical to saying "the vase can only become > > empty /at time t/, if there are balls removed /at time t/"; which is > > what it seems you actually mean. > > > > This doesn't follow from (1)..(8), which lack any explicit mention of > > what "becomes empty" means. However, we can easily make it an > > assumption: > > > > (T1) If, for some time t1 < t0, it is the case that the number of balls > > in the vase at any time t with t1 <= t < t0 is different than the > > number of balls at time t0, then balls are removed at time t0, or balls > > are added at time t0. > > > >> so if no balls are removed, the vase cannot become empty at noon. It was > >> not empty before noon, therefore it is not empty at noon. Nothing can > >> happen at noon, since that would involve a ball n such that 1/n=0. > > > > Now your logical argument is complete, assuming we also accept > > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > > (6), the number of balls changes at time 0; and therefore by (T1), > > balls are either placed or removed at time 0, implying by (5) and (6) > > that there is a natural number n such that -1/n = 0; which is absurd. > > Therefore, by reductio ad absurdum, the number of balls at time 0 > > cannot be 0. > > > > However, it does not follow that the number of balls in the vase is > > therefore any other natural number n, or even infinite, at time 0; > > because that would /equally/ require that the number of balls changes > > at time 0, and that in turn requires by (T1) that balls are either > > added or removed at time 0; and again by (5) or (6) this implies that > > there is a natural number n with -1/n = 0; which is absurd. So again, > > we get that any statement of the form "the number of balls at time 0 is > > (anything") must be false by reductio absurdum. > > > > So if we include (T1) as an assumption as well as (1)..(8), it follows > > logically that the number of balls in the vase at time 0 is not > > well-defined. > > > > Of course, we also find that by (1)..(8) and (T1), it /still/ follows > > logically that the number of balls in the vase at time t is 0; and this > > is a problem: we can prove two different and incompatible statements > > from the same set of assumptions > > > > So at least one of the assumptions (1)..(8) and (T1) must be discarded > > if we are to resolve this. What do you suggest? Which of (1)..(8) do > > you want discard to maintain (T1)? > > > > Cheers - Chas > > > > This is a very good question, Chas. Thanks. I'll have to think about it, > and I'm rather tired right now, but at first glance it seems like it > could be a sound analysis. I've cut and pasted for perusal when I'm > sharper tomorrow. > Here's some of my thoughts: When you say "noon doesn't occur"; I think "he doesn't accept (1): by a time t, we mean a real number t" When you say "if we always add more balls than we remove, the number of balls in the vase at time 0 is not 0", I think "he doesn't accept (8): if the numbers of balls in the vase is not 0, then there is a ball in the vase." When you say "an infinite number of balls are removed at time 0", I think "he does not agree with (6) if balls are removed at some time t, they are removed in accordance with the problem statement: i.e. there exists some natural number n s.t. n = -1/t and (some other stuff)". All these assertions follow a simgle theme: "If I require that my statemnents be /logically/ consistent, does the given problem make sense; and if so, what is a reasonable resonse?". Cheers - Chas
From: Ross A. Finlayson on 27 Oct 2006 00:31 Tony Orlow wrote: > > For what it's worth, and I know this doesn't add a lot of credibility to > Ross in your eyes, coming from me, but I think Ross has a genuine > intuition that isn't far off with respect to what's controversial in > modern math. Sure, he gets repetitive and I don't agree with everything > he says, but his cryptic "Well order the reals", which I actually > haven't seen too much of lately, is a direct reference to his EF > (Equivalence Function, yes?) between the naturals and the reals in > [0,1). The reals viewed as discrete infinitesimals map to the > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > answer your question, I think Ross makes some sense. But, of course, > coming from me, that probably doesn't mean much. :) > > TOE-Knee Hi, What is this IFR, "inverse function rule"? I've heard you mention it. Is it just general EF? Ross
From: David Marcus on 27 Oct 2006 00:34
cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: << snip >> > I think this "controversy" is in fact the difference between relying > utterly on inutition (as Ross seems to do) and relying utterly on > logical conclusions of some explicit set of assumptions (which is the > domain of what I would call "mathematics") which is at issue here. > > Just as logical conclusions from some set of assumptions can at times > be in conflict with our intuitions; so it is also true that we can hold > intuitions which are not logically compatible with each other. > > In my use of the word, mathematics is within that domain of discussion > which eschews the latter in favor of the former. It is a specialisation > of the domain of logic, rather than of the domain of physics. > > > Sure, he gets repetitive and I don't agree with everything > > he says, but his cryptic "Well order the reals", which I actually > > haven't seen too much of lately, is a direct reference to his EF > > (Equivalence Function, yes?) between the naturals and the reals in > > [0,1). The reals viewed as discrete infinitesimals map to the > > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > > answer your question, I think Ross makes some sense. > > Of course; there is nothing he says that is completely without /some/ > sort of sense. But I would say he is speaking /poetically/, not > mathematically; so in the context of sci.math, I can't respond to his > remarks. When mathematicians talk, they know which words have technical meanings and which don't. Ross simply uses the words without knowing the technical meanings. So, it gives the appearance of mathematics, but there is no actual communication of mathematical ideas. I once tried to teach some mathematics to some friends who weren't mathematicians via a weekly lunch seminar. We took a (fairly advanced) math book, and started reading it. When I read a math book, I can immediately categorize each sentence as definition, theorem, proof, or remark, even if the sentence isn't labeled as such. I was a bit surprised to discover that my friends weren't picking up on this at all. As such, they couldn't even begin to follow the book, since they couldn't tell what the purpose of each sentence was in the logical flow. They weren't even aware of the convention that a word in italics means the sentence is a definition of the word. -- David Marcus |