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

[Gareth McCaughan]

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.
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. :-)

> 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