An uncountable countable set
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From: Virgil on 11 Jul 2006 18:03 In article <1152639272.581323.205320(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Wrong. In order to determine the n-th line, one must count from 1 to n. > > > > Cantor does not claim to be able to provide any such list. > > And it is the provider of the list who must do that. Cantor only says > > that when someone provides him with that list, that he can find a > > number not listed. > > That is a contradiction, if the provider has listed all numbers, isn't > it? How is the provider to do that which has been shown impossible? When some provider actually has succeeded, only then is there a problem, but that has not happened, and cannot happen. So lets wait until it does before asking stupid questions about it.
From: Virgil on 11 Jul 2006 18:07 In article <1152639432.511655.220040(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > Except that after each (1,2), there will only be a finite initial > > subsequence in numerical order and an infinite terminal sequence not yet > > ordered. > > > > Using 1 origin indexing, the nth occurrence of (1,2) occurs at the > > (n^2 + n)/2 th position in the list of "transpostions". > > > > At which (n^2 + n)/2 th operation is the entire list ordered? > > At which number can I find the last element of Cantor's diagonal? The natural immediately following the one for the transposition that completes the simultaneous well-ordering and natural ordering of the rationals.
From: Virgil on 11 Jul 2006 18:10 In article <1152639512.932004.227390(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1152544884.364304.157620(a)75g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > > > >> > Either K is in the list (which would contradict analysis in the > > > > >> > same way as my original example), then there is an n with 10^(-n) = > > > > >> > 0. Or K is not in the list. > > > > >> > > > > >> K is not in the list. > > > > >> > > > > >> > Then there must be a position which cannot be enumerated by natural > > > > >> > numbers > > > > >> > > > > >> non sequitur. All positions are indexed by definition of decimal > > > > >> representation. > > > > > > > > > > All positions are indexed <==> K is in the list. It is a logical > > > > > equivalence for linear sets like my list. > > > > > > > > What kind of "logic" is employed here? > > > > > > That one which deserves its name, and which you unfortunately seem to > > > be not familiar with. The logic of a linear sequence like this sequence > > > of list numbers > > > > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > > "Mueckenh" insists that that sequence has a double meaning, both as > > unary representations of naturals and as binary, or other based, > > rationals, and requires that 0.111... satisfy both interpretations > > simultaneously. > > Ergo, it is garbage. > > You would call it bright if you were able to understand it. Garbage is never bright. I also choose not to try to understand why the sea is boiling hot and whether pigs have wings.
From: Virgil on 11 Jul 2006 18:12 In article <1152639642.547075.258360(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > All positions which can be enumerated are in the list, by definition. > > > Hence, the decimal representation of 1/9 does not exist. > > > > Then how is it that so many people use it? > > Because they do only believe they used it. For numbers, such things only exist by belief in them anyway, so if enough people believe in 0.111..., then it does exist.
From: Virgil on 11 Jul 2006 18:19
In article <1152639759.882706.252490(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > The first edge is mapped on the first path. > > > > Which of the infinitely many paths through that edge is the "first" one? > > > That at the root. All paths start at the root. > > > > > > > If this splits in two, 1/2 of the first edge in addition to the new > > > edge is mapped on the new paths. > > > > You are not allowed to split edges. Besides which, the first edge has > > already been entirely used up. > > It is inherited by the childs of the first path. Paths do not have children, only nodes have children. > > > > > > If the new paths split, 1/4 of the first edge and 1/2 of the > > > second and all of the third are mapped on each path. Do you know what a > > > geometric series is? What do you object to this mapping? > > > > If one can split edges into infinitely many peices one can equally split > > paths. it is only unsplit edges and unsplit paths that are to be matched > > up in injections, surjection or bijections.. > > You dislike fractions in calculations? Several thousand years ago > humans discovered that fraction can be useful. Why should they not be > useful in set theory? Some things are indivisible. An edge must start at one (parent) node and end at another (child) node. Any proper fraction of one won't reach. In the set of naturals {} is the first. What does half of {} mean? Or any other fractional part of {}? |