Re: Cantor's diagonal proof wrong?

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 11/22/04 

Date: Mon, 22 Nov 2004 16:00:05 GMT

In sci.math, fishfry
<BLOCKSPAMfishfry@your-mailbox.com>
 wrote
on Mon, 22 Nov 2004 06:08:02 GMT
<BLOCKSPAMfishfry-A1F0B0.22075621112004@netnews.comcast.net>:
> In article <20041122000411.328$H5@newsreader.com>,
> curt@kcwc.com (Curt Welch) wrote:
>
>> "*** T. Winter" <***.Winter@cwi.nl> wrote:
>> > In article <20041120000433.576$gt@newsreader.com> curt@kcwc.com (Curt
>> > Welch) writes:
>>
>> > But mathematics does not consider infinite processes either (not entirely
>> > true, see the halting problem).
>>
>> Right. I think I can write that better now.
>>
>> In math, once we produce a descripton of something, we just assume it has
>> always existed. Only our ability to write the description is new.
>> Anything we can write a description of, and which is consistent with the
>> axioms of the system we are working in is fair game to work with.
>>
>> However, much of what is done in math that way produces the same result
>> even if you do switch to a perspective that requires all new objects of the
>> description to be created by a process. It just requires you to look at it
>> in a slightly more complex way. You have to look at it as if you were
>> analizing the algorithm instead of the object itself. Cantor's diagonal
>> proof however doesn't work out the same if you use this non-mathematical
>> perspective.
>
> Does the set N of natural numbers exist?

No. Numbers exist only as concepts. Of course the set N can be
described abstractly (which is the best we can do about numbers
anyway); Peano in particular has a perfectly good set of axioms,
and a lot of ink (and, nowadays, electrons) has been used on
proving various things about said set, and other sets such as Q and R.

If we assume a list of reals, generated by a procedure of some sort,
we can assume that it is generating digits in an infinite 2-D
array by some method. If it's computable, Diag is computable;
it merely requires copying of one of the input parameters to the
other and then a modification of the output:

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

Diag can appear nowhere on the list, as the digit in the N'th place
is unequal for any entry List(N). However, any finite prefix of
Diag will appear in the list 1 out of 10^n times (where n is the
length of the prefix), if it truly is a list of scattered reals.
(I'm not sure how to phrase it better than that.)

However, equality of decimally-expressed real numbers is merely
semicomputable anyway; if they're equal the comparison never halts.
This may be a consequence of the infinite expansion, and the best
we can do is use the '...' (ellipsis) and make a leap of faith,
a leap that can land one in some bad territory, as one might assume
that e = 2.718281828... is rational, when it's not; a more accurate
representation is e = 2.7182818284590452... .

The best way out of this dilemma is to prove e's irrationality
by using another form, such as a series expansion:

e = sum(i=1,+oo) (1/i!)

as one cannot look at an infinite decimal very well. :-) 

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