An uncountable countable set
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From: Virgil on 30 Aug 2006 17:18 In article <1156363640.845840.187460(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The cardinal number aleph_0 is infinite while the ordinal number > remains finite in order to have infinitely many finite numbers. Nonsense. > > > For my part, I agree that the set of finite > > > naturals is finite, though unbounded, > > > > In that case you are not using standard mathematical terminology. I > > have no idea what a finite but unbounded set is. > > That's why you cannot understand mathematics. You fall back behind > Cantor. He knew it. Actually, Cantor knew better than that.
From: Virgil on 30 Aug 2006 17:20 In article <1156363950.351759.83230(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I do not see how I could avoid my conclusion. But if you are so sure > then give us at least one example how you completely index a number > without covering all the preceding numbers. Given the number 5, I index it with 1. In this process, I do not need to "cover" 2 or 3 or 4.
From: Virgil on 30 Aug 2006 17:22 In article <1156364007.547996.270100(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > According to Cantor: The infinite set of finite numbers. aleph_0 is > actually infinite, no natural number is actually infinite. Pretty good!
From: Virgil on 30 Aug 2006 17:23 In article <1156363768.777975.223810(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > It is the same with the staircase: If the total height is H, then there > > > must be at least one stair of height H. > > > > Wrong. Think asymptote. > > Now I understand your idea and your error. You intermingle the height > of the staircase and its least upper bound, which is not the same. As > long as you cannot find a stair that has height 1 you cannot assert > that the staircase had the height 1. Then an open interval from 0 to 1 cannot have length 1.
From: Dik T. Winter on 30 Aug 2006 19:11
In article <virgil-026B14.15185630082006(a)news.usenetmonster.com> Virgil <virgil(a)comcast.net> writes: > In article <1156363640.845840.187460(a)75g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > The cardinal number aleph_0 is infinite while the ordinal number > > remains finite in order to have infinitely many finite numbers. > > Nonsense. > > > > > For my part, I agree that the set of finite > > > > naturals is finite, though unbounded, > > > > > > In that case you are not using standard mathematical terminology. I > > > have no idea what a finite but unbounded set is. > > > > That's why you cannot understand mathematics. You fall back behind > > Cantor. He knew it. > > Actually, Cantor knew better than that. Actually Cantor made an error. It was in one of his first papers where he wrote that you also could sort of count with transfinite numbers, and you the numbers you could count where those of the same class. If I remember correctly (I do not have the book here at home) he made the error that you needed a number of a higher class when counting all numbers of a particular class. I will try to find it tomorrow (it is in the first article where he uses omega). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |