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

[David C. Ullrich]

On Sat, 28 Jul 2001 00:04:58 +0100, Gareth.McCaughan at pobox.com (Gareth
McCaughan) wrote:

>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.

It's the only one in standard use that I'm aware of.
I know several different "standard" "implementations"
of integers, rationals and reals as sets, but a different
implementation for natural numbers does not spring to
mind.

Of course infintiely many others are possible. But
what's another "implementation" of the _natural_
_number_ 2 as a set that's actually in general use?

Not that it matters.


>> >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.

Ah. No, I didn't mean to say that, and yes I suppose
that one might have got that idea. But if we're talking
about implementations _as_ _sets_, and implementations
that are actually _used_, then no I'm not aware of
another. What's an example?

(Hint: The two other "implementations" of natural numbers
that spring to mind are not implementations as sets, which
for some reason is what I thought the topic was. Those
would be regarding natural numbers as defined by Peano
arithmetic and regarding 2 as the class of all sets with
two elements (spelled out without using the word "two",
of course).)
 
>Gareth McCaughan  Gareth.McCaughan at pobox.com
>.sig under construc


David C. Ullrich