Date: Mar 10, 2013 4:35 PM
Author: Virgil
Subject: Re: Matheology � 222 Back to the roots

In article 
<53c0e229-49f3-496e-9de4-18c9f81edad1@14g2000vbr.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 9 Mrz., 23:53, William Hughes <wpihug...@gmail.com> wrote:
> > On Mar 9, 5:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >

> > > What do you understand by the not changeable function (d)?
> >
> > If x is a potentially infinite sequence of 0's and 1's
> > then we say (x) is associated to x, if (x) is an algorithm
> > which given a natural number produces a finite string
> > of 0's and 1's, such that for every natural number n,
> > (x) produces the nth FIS of x.

>
> Ok. So it is clear that a finite number has to be given and a finite
> string is produced.


That defines a functional relationship between the number and the
string, but does not limit the number of number-to-string pairings to
only a finite number of such pairings.
> >
> > We will say x is coFIS to (y) iff
> >
> >      i.  We have (x) associated to x and
> >          (y) associated to y
> >
> >      ii.  For every n, (x) and (y) produce the same
> >           finite string.

>
> "Every given n" is tantamount to "there is a last given n".


Not outside Wolkenmuekenheim!

> This maximum is the same for line l_max and max FIS of d.

Not outside Wolkenmuekenheim!
Only inside Wolkenmuekenheim can one have natural numbers without
successors.

> >
> > Some comments.
> >
> > We can never have more than a limited number of the strings
> > produced by (x) existing at any time, since we can never
> > have more that a limited number of natural numbers.

>
> Correct.

> >
> > We cannot show that x is coFIS to (y) by using (x)
> > and (y) to produce "all possible strings" and
> > comparing them, since "all possible natural numbers"
> > does not exist.  However, we may be able to use induction
> > to show "For every n, (x) and (y) produce the same finite
> > string"

>
> Induction is fine, but also restricted to the (variable) maximum.


Requiring the existence of a natural number with no successor!
> >
> > To show that x is coFIS to (y), it is not enough
> > to show that  every existing FIS of x, is equal to an
> > existing FIS of y.

>
> More cannot be shown.


Then such coFISm cannot be shown at all, at lest inside
Wolkenmuekenheim.
> >
> > d, the diagonal of the list L, is a potentially infinite
> > sequence of 0's and 1's with associated (d):
> > for any natural number n produce
> > a sequence of n ones.

>
> This sequence is identical to a line l_max of the list L, by
> construction of d_max.


But outside of Wolkenmuekenheim there is no such alleged d_max possible,
as every member of d has a successor.
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