Re: A simple question about integers

From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 09/15/04 

Date: 14 Sep 2004 19:01:49 -0700

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.

I suspected that something like this would happen, and that is why I
asked of the consequences of accepting such an awkward thing
(elsewhere on the thread). This was precisely the kind of answer that
I was looking for, I was not trolling, but because of the constant
rush of trolls on this group and an unfortunate resemblance to the
subject of another thread perhaps, most people mistook me as a troll.
Thank you for your post.

Carry should not pose a problem for addition operation over two
infinite binary strings intuitively, (because of the metamathematical
reason that each step of computation is still finite, etc.) but you
have an excellent point, namely that the result of arithmetic
operations do not look sensible, however way we define them. When both
A and B have infinitely many digits, their addition is still in this
larger structure (so basic algebraic properties do seem to hold, etc.)
but the resulting larger structure is *so* inelegant, and we would
prefer the current elegant definition of Z over the inelegant
structure of these strange large whole numbers... That's a really good
answer, and has much intuitive appeal. When I meet my friend the next
time (probably a reincarnation of Kronecker, he liked to challenge
Cantor's mathematics so much!), I will try this argument. :)

I think I could also explain why the results are insensible like this,
a nice property of Z is that we can always compare two of its members
reliably. How to do that with this larger structure? We would fail,
because there is no "largest digit" that allows us to even start
comparing in an orderly fashion. These "large whole numbers" are
shattered! We can't even compare these two guys like we compare the
proper integers of Z, this creates a dichotomy already, and hence even
if such a larger structure is to be treated, it should be treated as
*separate* from Z.

I do think that when confronted with the question my friend asked, we
need to give a stronger argument like this one than the short
answer...

Thanks again.

Best Regards, 

--
Eray Ozkural

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