An uncountable countable set
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From: Alan Morgan on 13 Oct 2006 17:18 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. It's arithmetic. You can rearrange the terms. That's the whole problem. You can make it sum up to (almost) anything you like and that includes zero. 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 -- Defendit numerus
From: David Marcus on 13 Oct 2006 20:55 Tony Orlow wrote: > Han de Bruijn wrote: > > Virgil wrote: > > > >> http://en.wikipedia.org/wiki/ZFC > >> 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). > > So, we can interpret the empty set as 0, the origin, and then define > successor any way we want. IF we define the successor of n as n+1, then > we get the naturals. If we define the successor as 1-1/2(1-n), then we > get our Zeno moments. The inductive set produced depends on what the > null set represents and how successor is defined. No. In the axiom of infinity, S(x) is defined as x union {x}. However, the details of axiomatic set theory aren't really relevant to the present discussion. -- David Marcus
From: David Marcus on 13 Oct 2006 20:58 Han de Bruijn wrote: > Virgil wrote: > > > In article <66a8$452f4298$82a1e228$30886(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>Randy Poe wrote about the Balls in a Vase problem: > >> > >>>Tony Orlow wrote: > >> > >>>>Specifically, that for every ball removed, 10 are inserted. > >>> > >>>All of which are eventually removed. Every single one. > >> > >>All of which are eventually inserted. Every single one. > > > > None are reinserted after being removed but,each is removed after having > > been inserted, so that leaves them all outside the vase at noon. > > Huh! Then reverse the process: first remove 1, then insert 10. It must > be no problem in your "counter intuitive" mathematics to start with -1 > balls in that vase. Consider this situation: Start with an empty vase. Add a ball at time 5. Remove it at time 6. How you would translate that into Mathematics? -- David Marcus
From: David Marcus on 13 Oct 2006 20:59 Han de Bruijn wrote: > Virgil wrote: > > > In article <9020$452f46c4$82a1e228$31963(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>Virgil wrote about the Balls in a Vase problem: > >> > >>>Everything takes place before noon, so that by noon, it is all over and > >>>done with. > >> > >>Noon is never reached, because your concept of time is a fake. > > > > No one expects the experiment to take place anywhere except in the > > imagination, so that everything about it, including its time, is > > imaginary, but logic continues to hold even there, at least for > > mathematicians. And logic says that a ball removed from a vase is not > > later in the vase. > > Since your logic and the logic of others give contradictory results for > the same problem, logic alone is unreliable. Are you saying that Mathematics gives contradictory results for a problem? If so, please state the problem. -- David Marcus
From: David R Tribble on 13 Oct 2006 21:48
Tony Orlow wrote: >> No, the inductive proof of an equality applies to all n, finite or >> infinite. But "is finite" is an inequality, equivalent to "<oo". >> lim(n->oo: n)=oo, not <oo. You can only increment a finite value a >> finite number of times before you get infinite values out of it. > David R Tribble wrote: >> How many times? > Tony Orlow wrote: > Less than any infinite number of times. So now you're saying that a finite value can be incremented a finite number of times (any number less than an infinite number of times) and you'll get an infinite value? Before you said that an infinite value results when you increment a finite value an infinite number of times. And here I was thinking all this time that any finite value plus another finite value always resulted in another finite value. Where, oh where, did I go wrong? |