Re: induction vs Cantor II

From: Daryl McCullough (daryl_at_atc-nycorp.com)
Date: 11/29/04 

Date: 28 Nov 2004 19:16:46 -0800

Poker Joker says...
>
>Sorry, forgot sci.math
>
>> Summary from the first thread:
>>
>> L_1 is a list of reals. This creates an implied
>> mapping (injection) F_1 from the naturals to the reals.
>>
>> D_n is the Cantor number, CN (anti-diagonal number),
>> that can be formed using the mapping F_n.
>>
>> Let L_n+1 be a list of reals by inserting D_n into L_n
>> at row 2n.
>> Row i in L_n+1 = Row i in L_n if i < 2n.
>> Row i in L_n+1 = Row i-1 in L_n if i > 2n.

Okay. So the original list L_1 is

   r_1
   r_2
   r_3
    .
    .
    .

L_2 is the list (where d_1 is the anti-diagonal number of L_1)

   r_1
   d_1
   r_2
   r_3
    .
    .
    .

L_3 is the list (where d_2 is the anti-diagonal number of L_2)

   r_1
   d_1
   r_2
   d_2
   r_3
    .
    .
    .

In the limit, we have L_infinity which will look like the following

   r_1
   d_1
   r_2
   d_2
   r_3
   d_3
   r_4
   d_4
   r_5
   d_5
    .
    .
    .

This list contains all the elements of L_1 plus all the
anti-diagonal numbers d_1, d_2, etc. If you use Cantor's
diagonalization procedure on *this* list, you get a new
number d_infinity that is not equal to r_1, nor d_1, nor
r_2, nor d_2, etc.

d_infinity is guaranteed to be different from r_1 in the
first decimal place, different from d_1 in the second
decimal place, different from r_2 in the third place,
different from d_2 in the fourth place, etc. 

--
Daryl McCullough
Ithaca, NY

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