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Topic: This Is *PROOF* that AD never produces a New Digit Sequence!
Replies: 3   Last Post: Jan 2, 2013 7:19 PM

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Graham Cooper

Posts: 4,227
Registered: 5/20/10
Re: This Is *PROOF* that AD never produces a New Digit Sequence!
Posted: Jan 2, 2013 6:59 PM
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On Jan 3, 7:32 am, George Greene <gree...@email.unc.edu> wrote:
> On Dec 29 2012, 5:18 pm, Graham Cooper <grahamcoop...@gmail.com>
> wrote:
>

> > AD METHOD (binary version)
> >   Choose the number 0.a_1a_2a_3...., where a_i = 1 if the i-th
> >   number in your list had zero in its i-position, a_i = 0 otherwise.

>
> > LIST
> >   R1= < <314><15><926><535><8979><323> ... >
> >   R2= < <27><18281828><459045><235360> ... >
> >   R3= < <333><333><333><333><333><333> ... >
> >   R4= < <888888888888888888888><8><88> ... >
> >   R5= < <0123456789><0123456789><01234 ... >
> >   R6= < <1><414><21356><2373095><0488> ... >
> > ....

>
> Well, YES, we KNOW that this is the kind of thing (a square wxw list)
> that we are talking about.
>

> > By breaking each infinite expansion into arbitrary finite length
> > segments

>
> The word "arbitrary" DOES NOT HELP you here.
> If it TRULY is "arbitrary" then I can arbitrarily pick ONE as my
> segment-length.  And then the whole notion of segments JUST GOES AWAY.
> It's JUST DIGITS.  Your going to segments DOESN'T CHANGE anything.
> If there were originally 10 digits (0-9) and you go to length-3
> segments,
> then that is JUST LIKE having A THOUSAND digits an STAYING with length-
> *1*
> segments, i.e., NOT HAVING ANY SEGMENTS AT ALL.
>
> Dumbass.


*I* provide the List of All Reals.

*I* guarantee all possible segments are listed, in every possible
order.

PROOF BY INDUCTION.


BASE STEP
A(n) All 10^n segments occur in segment position 1, 2, 3,...
infinitely many times each in each position.


INDUCTIVE STEP
If <s1> <s2> are 2 consecutive listed segments in BASE STEP
< s11 s12 s13... s1m s21 s22 s23 .. s2n>

where m is the length of segment 1
and n is the length of segment 2

is ALSO a segment of the BASE STEP

Therefore by Induction, all sequences of all segments are listed.

GIVEN SUCH A LIST IN SEGMENTED FORM

LIST
R1= < <314><15><926><535><8979><323> ... >
R2= < <27><18281828><459045><235360> ... >
R3= < <333><333><333><333><333><333> ... >
R4= < <888888888888888888888><8><88> ... >
R5= < <0123456789><0123456789><01234 ... >
R6= < <1><414><21356><2373095><0488> ... >
....


You cannot calculate a missing anti-diagonal in *any* segmented form.



Herc




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