```Date: Apr 5, 2013 4:40 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224

On 5 Apr., 21:03, William Hughes <wpihug...@gmail.com> wrote:> On Apr 5, 6:04 pm, WM <mueck...@rz.fh-augsburg.de> wrote:>> > On 5 Apr., 12:08, William Hughes <wpihug...@gmail.com> wrote:>> <snip>>> > > There is an infinite set of lines D> > > such that any finite subset of D can be removed.>> > What has to remain?>> This depends on the finite subset removed.> If the finite set removed is E then> D\E has to remain.  Note that whatever> subset E is chosen the number of lines> in D\E is infiniteHow do you call a set E the number of elements exceeds any givennatural number? (You do not claim that we can only remove a set E withless than a given natural number, do you?)>  (but of course we> do not know which lines are in D\E).How do we call a set when we cannot biject it with a FIS on |N?> However, D cannot be removed without> changing the union of the remaining lines.That is correct, if D is not more than the union of all finite lines.But if so, then every Cantor list that contains all rational numbershas the following property:For every n in |N: There are infinitely many lines that have the samefinite initials sequence d_1, d_2, d_3, ..., d_n of digits as the anti-diagonal.Only if the diagonal is more than every FIS, i.e., the list is morethan every finite lines, then this proof could be objected. For all nin |N in is valid.So you are caught in a circulus vitiosus: Either actual infity D ismore than all its FISs, then all can be removed without changing whichis obviously nonsense, or D is not more, then there is a proof againstCantor's theorem which is as valid as Cantor's proof.Regards, WM
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