Re: A simple question about integers

From: Gottfried Helms (helms_at_uni-kassel.de)
Date: 09/15/04 

Date: Wed, 15 Sep 2004 08:31:08 +0200

Am 15.09.04 04:01 schrieb Eray Ozkural exa:
> Angus Rodgers <angus_prune@bigfoot.com> wrote in message news:<5lvdk0lnnki8hpsrlcj2ajfifltom9uud2@4ax.com>...
>
>>The long answer is also `no'. What I mean is that even
>>if you try to embed Z into a larger structure which has
>>some of the same properties as Z, but which also contains
>>`numbers' with infinite decimal expansions, such as:
>>
>> ...57803170965
>>
>>- leaving aside the question of what this would `mean' -
>>
>>then you are not going to be able to define addition in
>>your new structure in any way resembling addition in Z:
>>because in a case where there is no `carrying' you will
>>certainly want to perform addition digitwise (e.g. in Z,
>>320 + 539 = 859 because 0 + 9 = 9, 2 + 3 = 5, 3 + 5 = 8),
>>but this forces you to define the sum of the above number
>>with the following (equally reasonable-looking) `number':
>>
>> ...42196829034
>>
>>as:
>>
>> ...99999999999
>>
>>and you will not even be able to add 1 to that `number'
>>in any sensible-looking way.
>>
>>I don't see any way around this. Do you?
>
>
> I don't either.
>
There are two more problems with that:
 such a definition, written down or not, introduces the
 concept of divergent series, where we know, that results
 may be arbitrary depending simply on the order of operations.
 Now such a divergent series (a1 + 10*a2 + 100*a3+...) may
 be lesse difficult in that regard, since we have a fixed index
 which at least fixes that order problem.

 the more serious problem in that way is, that with arbitray
 sequence of digits we even cannnot define a greater/smaller
 order between two such numbers... since they have no
 highest digit. Since the greater/smaller-relation is also
 basic for the applicability of our mathematical operations
 (and their reflection to empirical matters which makes math
 meaningful) this construction seem to be completely useless
 in math.
 The only reason I could vaguely speculate about things like
 that could be if one finds out a relation to the sometimes
 helpful concept in analytic geometry with its "infinite far
 point". I'm not able to say much more about that, but I
 assume that even for such a thing - if it would be helpful -
 only the ...999-version of that concept could come in
 charge.

Gottfried Helms