Date: Feb 4, 2013 3:21 AM
Subject: Re: Matheology § 203
On 3 Feb., 23:09, William Hughes <wpihug...@gmail.com> wrote:
> On Feb 3, 10:58 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 3 Feb., 22:29, William Hughes <wpihug...@gmail.com> wrote:
> > > > > We can say "every line has the property that it
> > > > > does not contain every initial segment of s"
> > > > > There is no need to use the concept "all".
> > > > Yes, and this is the only sensible way to treat infinity.
> > > So now we have a way of saying
> > > s is not a line of L
> > > e.g. 0.111... is not a line of
> > > 0.1000...
> > > 0.11000...
> > > 0.111000....
> > > ...
> > > because every line, l(n), has the property that
> > > l(n) does not contain every initial
> > > segment of 0.111...
> > But that does not exclude s from being in the list.
> It certainly excludes 0.111... from being a single line
> of the list.
> So the question is now
> Can a potentially infinite list
> of potentially infinite 0/1
> sequences have the property that
> if s is a potentially infinite 0/1
> sequence, then there is a line, g, of L
> with the property that every
> initial segment of s is contained in g
Of course, every FIS is in a line. And every FIS contains all
contains every FIS of 0.111... as well in the (unchanged) diagonal as
in a line. Obviously the diagonal cannot have more 1' than every line.
(Of course you cannot expect to define 0.111... without a last element
1_omega but to find a last line g.)