[OT] Number theory [Was: A use for integer quotients]

[Martin Sjögren]

On Thu, Jul 26, 2001 at 12:01:15AM +1000, michael wrote:
> > But strictly
> > speaking the Integer 2 and the Real Number 2 are different entities.
> 
> So strictly speaking, Z (the set of integers) is not a subset of R (the set
> of reals)?

Hmm. Strictly strictly strictly I don't think so.

It's all about how you define the sets of course, and I guess that can be
done in a number of ways.

The numeral 2 is the successor of the successor of zero, that is S(S(0)),
which in set theory usually is {\emptyset, {\emptyset}} (0 = \emptyset and
S(x) = {x} U x).

That's N, given this, you usually define elements in Z as (n,m) where n
and m in N (ouch, case sensitivity problem there! <n/2 wink>) and let the
negative number "-3" be (0,3).

Now you can define the elements of Q as (p,q) where one of them is in Z
and the other in N, (and q!=0).

The real numbers are usually defined with Dedekind cuts (is that the
correct English term?) which is a bit complicated.

If you define it this way, then sure, Z isn't a subset of R...

</mathetmatical digression>

Martin

-- 
Martin Sjögren
  martin at strakt.com              ICQ : 41245059
  Phone: +46 (0)31 405242        Cell: +46 (0)739 169191
  GPG key: http://www.strakt.com/~martin/gpg.html