An uncountable countable set
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From: Han de Bruijn on 13 Oct 2006 03:56 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. Han de Bruijn
From: Virgil on 13 Oct 2006 04:00 In article <e5389$452f434a$82a1e228$30886(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> 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. > > It was recognized long before Cantor that there isn't an answer at all > to a meaningless question. > > Han de Bruijn But there was then, and remains today, wide disagreement on exactly which questions are meaningless. And it is not an issue on which I would trust HdB's judgement.
From: Virgil on 13 Oct 2006 04:09 In article <bc2e$452f465d$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: > > > The point is that after a ball is removed from the vase it is no longer > > in the vase, and every ball is removed before noon. > > The point is that after a ball is removed from the vase, ten others are > inserted. And every ball is inserted before noon. And each ball inserted before noon is also removed before noon. Those who fail to see this are not keeping their eyes on the ball. > > Because different, but valid, logics lead to contradictory conclusions, > the end result is undecidable (: just take a look at the length of this > Balls in a Vase debate ...) The end result is decidable if one does not allow irrelevancies to obscure the fact that each ball is inserted and removed before noon. What happens with other balls does not affect this pertinent fact. What bugs many people is that they cannot imagine discontinuous behavior in the physical world, so are unprepared to imagine it in the imaginary world in which this gedankenexperiment must take place.
From: Virgil on 13 Oct 2006 04:13 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.
From: Ross A. Finlayson on 13 Oct 2006 05:05
Han de Bruijn wrote: > Mike Kelly wrote about the Balls in a Vase problem: > > > Ah, but noon is not a part of the sequence of iterations. No more than > > 0 is an element of the sequence 1, 1/2, 1/4, 1/8, .... > > Thus the question is whether the sequence (number of balls) converges. > > > The question asks how many balls are in the vase at noon. Not at some > > iteration. > > Well, it does not converge. So this question of yours is meaningless. > > Han de Bruijn Hi Han, How are you today? Wie geht es Ihnen? Ross |