An uncountable countable set
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From: Tony Orlow on 13 Oct 2006 22:05 Alan Morgan wrote: > In article <452ef7a0(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: >> Alan Morgan wrote: >>> In article <452e8c2a(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> What is sum(n=1->oo: 9)? >>> I think you actually mean, what is 10-1+10-1+10-1.... >>> >>> It was recognized long before Cantor that there isn't a simple answer to >>> that question. >>> >>> Alan >> There is if you prohibit rearranging the terms to change the relative >> frequencies of the two terms. Group all you like without rearranging. >> This series is (+10-1)+(10-1)+(10-1)+... > > Yes, but no one has ever said that you can't rearrange the terms. I'm sure many have tried to formulate the circumstances where it violates reason. The current understanding is that you can do whatever you want. Go ahead, and see where it gets you. Where there is a specific sequence of events specified, violating that sequence makes it different situation. It's > arithmetic. You can rearrange the terms. You cannot rearrange the sequence of events, where the sequence is explicitly stated in the problem. +10-1, repeat. That's the whole problem. You > can make it sum up to (almost) anything you like and that includes zero. Only by artificially rearranging the sequence, and "shoving" all the 10's down the line while pretending the '-1's' make up for them. I call this the "chop 9 of the little children's fingers and make them use the tenth one to pass them down to the "end"' syndrome. > I'm not claiming that every interpretation of this infinite sum yields > zero, but you (and others) are claiming that the only possible answer is > that it sums to infinity and we've known since long before Cantor than > anyone who claims there is only one correct answer to that sum is full > of it. > > Alan What are you full of? Balls? Whatever. Without rearranging the terms, there is one sensible way to group them, and get a simple formula for a sum, which, by the way, diverges. When ball n is removed, balls n+1 through 10n remain.
From: Tony Orlow on 13 Oct 2006 22:08 Virgil wrote: > In article <452fc7c2(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> I know that at no time have all the balls previous inserted been >> removed, but only 1/9th of them, since 1 is removed for every 10 >> inserted. What is the flaw in that logic? > > At no time BEFORE NOON have all the balls been removed. > Since in the original statement of the problem, one was given the > precise time before noon for removal of every ball that was to be > inserted, by noon it does not matter a whit how long the ball was the > vase. Hi Virgil. Wow! It's really nice to see you! I can't believe you actually stooped so low as to respond to one of my little comments. Thank you, Sir. I really appreciate what little attention I get. :) So, what were you saying? Oh yeah, that all balls were in the vase before noon. But that's not true.
From: Tony Orlow on 13 Oct 2006 22:10 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.
From: MoeBlee on 13 Oct 2006 22:12 Tony Orlow wrote: > The Axiom of Infinity employs ordinals - a > big mistake. What do you mean by 'employs'? We do not even have to have defined 'ordinal' to state the axiom of infinity. The axiom of infinity makes no mention of the predicate 'is an ordinal'. Of course, 0 is an ordinal, and the successor of an ordinal is an ordinal, but we do not need to "employ" those facts just to state the axiom of infinity. MoeBlee
From: MoeBlee on 13 Oct 2006 22:13
Tony Orlow wrote: > However, > > the details of axiomatic set theory aren't really relevant to the > > present discussion. > > > > Indeed they are. Then one would think you'd learn those details. MoeBlee |