In article <firstname.lastname@example.org>, WM <email@example.com> 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. --