Re: November 25 is Infinite Clause day!!

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 11/21/04 

Date: Sun, 21 Nov 2004 18:00:06 GMT

In sci.logic, David Bernier
<david250@videotron.ca>
 wrote
on Sat, 20 Nov 2004 23:23:55 -0500
<vLUnd.98560$De5.1797114@wagner.videotron.net>:
> HERC777 wrote:
>> Merry X-count-mas!
>>
>> Do you believe in Infinite Clause? Will he make it to your real place
>> in time and all of the infinite places and make something new for you?
>>
>> Have you been good and read your maths text so you too can see
>> Infinite Clause? He's keeping a list and he's checking it twice!
>>
>> CANTOR :
>> The uncountable number does not have the 1st digit of the 1st
>> countable number.
>>
>> UTM(neN,d) :
>> ALL 10 digits are present in the 1st digit place an infinite number of
>> times, on the list of computable reals.
>>
>> CANTOR :
>> The uncountable number does not have the 2nd digit of the 2nd
>> countable number.
>>
>> UTM(neN,d) :
>> ALL 10 digits are present in the 2nd digit position (following every
>> possibility of the 1st digit) for an infinite number of reals.
>>
>> CANTOR :
>> The uncountable number does not have the 3rd digit of the 3rd
>> countable number.
>> ...
>>
>>
>> Start looking at reality sci.math not your text, not David Ullrich and
>> Barb Knox who *make money teaching texts*. 0.123... is on the list of
>> computable numbers infinite times with infinite possible tails after
>> the 123. 0.654... is there infinite times. ALL PREFIXES of UNLIMITED
>> LENGTH are ON THE LIST. You ----=== C A N N O T ===--- generate a
>> new sequence of digits that is not on the list of computable reals.
>> ALL PERMUTATIONS ARE THERE. The diag number must be SOME sequence of
>> digits, like 0.123..., like 0.654... ITS NOT A NEW NUMBER, the elves
>> went on strike!
>
> [snip]
>
> Where is THE LIST?
>
> David Bernier

Not to mention HERC777's understanding. The question is not whether
the diagonal number's digits are within the list (most likely,
they are, unless "The List" consists of, say, numbers exclusively
in the Cantor set and the digits are taken from base 3 instead
of base 10), but whether the number itself is on the list.

For purposes of this post, I'll posit that the number is
represented Diag, and each digit can be fetched from a
function Diag(N), and digit N of entry M in The List can
be represented List(M,N); the M'th entry in The List can
be represented List(M). (For technical reasons no entry
on the List can have an infinite trail of 9's.)

Now List can be computed by a variety of methods:
- a Turing machine, spitting out numbers.
- a wizard's ball.
- a function.
- enumeration of a countable set, such as the rather simple one T_10 =
  {.0} union {k/10^n; k > 0, k, n in J, (k mod 10) != 0, 10^(n-1) < k < 10^n)}
  which contains all finite decimal expansions in [0,1)

It doesn't really matter.

We also need to construct Diag. This is fairly simple,
and one can have many variants, so we need just pick one:

if List(N,N) = 4 then Diag(N) = 5 else Diag(N) = 4

The question is whether Diag is in the List or not. This question
is too vague in some respects; a more formal version might be

Does an N exist such that Diag = List(N)?

and that needs improvement too; the final version is

Does an N exist such that for every positive M in J,
   Diag(M) = List(N,M)?

The answer clearly is no (if M=N Diag(N) != List(N,N) by construction).

It's also clear that T_10 contains every digit an infinite number of times,
in any digit slot one cares to specify. T_10 is also dense;
Diag may not be on the list but T_10 will contain numbers arbitrarily
close to Diag. However, Diag is simply not in T_10, in the same
manner that 1/3 is not in the set { (10^n - 1) / (3 * 10^n): n in J, n > 0}
= {.3, .33, .333, .3333, ...}, though again one can find elements
that are arbitrarily close to 1/3.

Cantor's first proof also addresses the issue, in a more abstract fashion.

If one wants to rescope the problem, one can state Diag(List) instead
of Diag and propagate forward as necessary; the English equivalent
might be

   "for *any* List of reals, Diag is not on that particular List"

which might work even better, though HERC777 has tried to insert Diag
into the postulated List (which won't work as it changes the List). 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.

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