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

[David C. Ullrich]

On Thu, 26 Jul 2001 22:14:30 +0100, Gareth.McCaughan at pobox.com (Gareth
McCaughan) wrote:

>David Ullrich wrote:
>
>[someone else:]
>>> So strictly speaking, Z (the set of integers) is not a subset of R (the set
>>> of reals)?
>> 
>> Strictly speaking yes, but nobody _ever_ speaks this strictly
>> (except in contexts like the present).
>> 
>> We math guys have reduced more or less everything to sets these
>> days - fewer things at the bottom to keep track of. So when we
>> want to define a "type" with certain "properties" we figure out
>> how to "code" such a thing as a certain sort of set. But almost
>> always what set something actually is makes no difference, all
>> that matters is how it works.
>> 
>> 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.

>That's one way to define the number 2, which happens to be the
>usual one at the moment. It's not the only way. Frege wanted
>2 to be the set of *all* unordered pairs {x,y}, and there are
>versions of set theory in which that definition works. In NF,
>sometimes 2 is defined as {{{}}}. Or there's the Conway approach,
>according to which the primitive notion is that of "ordered
>pair of sets" and 2 is an equivalence class whose "typical"
>member is ({0,1}, {}).
>
>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.

>> So yes, strictly speaking the integers are not a subset
>> of the reals. But nonetheless if someone asks "Are the
>> integers a subset of the reals?" the _right_ answer
>> is "yes"; the answer becomes "no" only if we're focussing
>> on the irrelevant aspects of things.
>
>Right. And, actually, I'd say that the isomorphic image
>of (say) Z inside R *is* the integers. For exactly the
>reasons you mention, what you really mean when you say
>"the integers" is "anything that behaves like the integers".
>That copy of Z is such a thing. The fact that for some
>foundational purposes you start by constructing another
>copy of Z doesn't invalidate that. So, I say: for *all*
>purposes "the integers are a subset of the reals", unless
>for some technical reason you *have* to pick an implementation
>of the integers *and* can't make it be the one that lives
>inside R.
>
>-- 
>Gareth McCaughan  Gareth.McCaughan at pobox.com
>.sig under construc


David C. Ullrich