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

[Gareth McCaughan]

David C. Ullrich wrote:

> >> Strictly speaking the natural number 2 is the set {{},{{}}}
> >
> >Tut. You're failing to distinguish interface from implementation.
> 
> Tut what? If you read more than one paragraph you see that my
> point was exactly that when we say that Z is not strictly a
> subset of R that's an irrelevant implemenation detail.

You said "Strictly speaking the natural number 2 is ..."
and gave one possible implementation. It's by no means
the only one, as I'm sure you know; but you didn't say
so.

> >I'm pretty sure you already know all this, but it's worth
> >saying explicitly. :-)
> 
> Probably the fact that I _did_ say explicitly that there
> are several different "standard" ways to implement integers
> and reals is what makes you suspect I know this, eh? Very
> astute of you.

I appear to have given offence, and I'm very sorry for
that. What I actually said is that because you said explicitly
that there are several ways to implement integers and reals,
you almost certainly are well aware that there are also
several ways to implement natural numbers.

I was just concerned that some people might conceivably
misunderstand you to be saying, as Kronecker didn't :-),
that God made the von Neumann implementation of the
natural numbers and all else is the work of man.

-- 
Gareth McCaughan  Gareth.McCaughan at pobox.com
.sig under construc