An uncountable countable set
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From: Dik T. Winter on 17 Aug 2006 11:04 In article <1155818644.699954.82400(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <J42BHx.IzE(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > > > In article <1155664930.866986.157410(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > > What I do not understand is how you can believe that someone (except, > > > > perhaps, Virgil) could share your opinon that only in one dimension the > > > > infinity was actually realized, but not in the other. > > > > > > Just because there is no last line. If there were a last line, it would > > > have width aleph-0, but there is no last line. > > > > Let's have an easier example. The blocks are 1/2^n high and 1-1/2^n wide. > > I propose the following stair > 1 - 1/2^n high and 1 - 1/2^n wide > up to the n-th step, in order to have symmetry. I propose not (and I did not use those values). > > When you make a stair of them, when you complete, you have a stair with > > height 1 and width 1. But there is neither a block that is 1 wide, nor > > a block that is at height 1. On the other hand, for every positive k, > > there are blocks that are beyond the height of 1-k and beyond the width > > of 1-k. Both in height and in width the blocks just do not reach the > > boundary line. And note that in both "infinity is reached". > > What you argue is a potential infinity. Infinity, aleph_0, here reduced > to 1, is present, actually existing, according to Cantor, in width. All > steps do exist. But, according to him, infinity, here 1, is not > present in height. It is, also according to him. Both in width and in height. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 17 Aug 2006 15:13 In article <1155817776.184870.134560(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The list of sequences of 1's > > 1 0.1 > 2 0.11 > 3 0.111 > ... > n 0.111...1 > n+1 0.111...11 > ... > > is constructed such that the segment from 1 to n does not contain > numbers which can completely index or cover the digit positions of a > number which does not belong to the segment, like the sequence n+1 and > all sequences which are larger, i.e., which have more 1's. > > As 0.111... does not belong to the list, it does not belong to any > segment of the list. Hence its digit positions cannot be completely > indexed or covered by the sequences of the list. 1 --> 0.111... 11--> 0.1 111 ---> 0.11 .... indexes them all. > > Now you may claim, that the mathematics of finity does not hold in > infinity, but that there everything is different. But that is not > mathematics in my opinion. We have often seem how little "Mueckenh"'s opinion is worth in mathematics. > May be that it is mathematics in some one > other's opinion. Nevertheless it is not useful to continue the > discussion on this topic between us. Then stop. > That is, in my opinion, purest nonsense. But the believer will never > give up his belief. "Mueckenh" is the believer here, he speaks for himself alone. "Mueckenh" claims some profound universal truth which he cannot justify except by faith. We only claim that what we can prove follows from certain axioms follows from those axioms.
From: Virgil on 17 Aug 2006 15:20 In article <1155817888.248664.91020(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > Nobody ever has said that the diagonal is ever completed. There is a > > > > definition of that number, and it is easy to show, with that > > > > definition, > > > > that that number (when completed) would be different from all numbers > > > > on > > > > the list. > > > > > > But if it is impossible to complete it, then the result is void. > > > > There are many real numbers that can not be completed. But by the > > definition of "real number", each sequence of decimal digits defines > > a real number, and it is easily shown, by the above, that that real > > number is not in the list because it is different from each number > > in the list. > > > Oh no. It is not different for any infinite line but only for some > finite lines. Which numbers in the list is it NOT different from? We can show it is different from the first. We can show that if it is different from any one of them, it is also different from the next one. By the inductive principle, which is valid in our axiom systems, it is thus different from each and every single one of them. Which numbers in the list does "Mueckenh" claim it is NOT different from? > If you tell me what is to be understood by "set" and by "existence" in > set theory. They are among the undefined primitives of set theory. I do not know your opinion about that matter. Zermelo > refrained from defining what a set is. So do we, at least in any formal sense. > > Regards, WM
From: Virgil on 17 Aug 2006 15:22 In article <1155817978.033024.136310(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > These positions are all in the list - by definition. > 0.111... is not in the list by mathematical proof. But it is in the list 0.111... 0.1 0.11 0.111 ....
From: Virgil on 17 Aug 2006 15:26
In article <1155818167.396269.42480(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > omega / omega = 1 because Cantor stated that it is a fixed quantum and, > therefore, omega = omega. Cantor never defined any "division" for ordinals or cardinals in general, so in Cantorean mathematics "omega/omega" has no meaning. So precisely what does "Mueckenh" mean by "omega/omega" ? |