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

[Gareth McCaughan]

David Ullrich wrote:

[I said:]
>> 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.

It's much the commonest, mostly because it appears to be
the simplest way to do it within ZF and most mathematicians
think that set theory == ZF (or perhaps ZFC). :-)

> 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?

Depends, I suppose, on what you mean by "in general
use". I'd say that *none* of them is in general use,
in the sense that no one ever depends on the details
when doing real mathematics. The von Neumann construction
is certainly far commoner than any other I know of in
textbooks on the foundations of mathematics.

However: Randall Holmes's book "Naive set theory with
a universal set" (which uses NFU as its set theory)
defines natural numbers as equivalence classes modulo
having-the-same-size, so that 2 is

  { {x,y} : x != y }

and so on. This was Frege's original definition too,
though he didn't have a consistent account of set theory
to fit it into.

> Not that it matters.

Indeed.

>> 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?

The one I gave above is an example. Or you can say that
0 = {} and n+! = {n}. That was Zermelo's definition,
15 years before von Neumann's, and it's the one adopted
in Quine's book "Set theory and its logic" (though he
is at pains to make it clear that there are other
options).

I don't know whether you'd consider Conway's approach
to be "_as_ _sets_"; Conway himself prefers to regard
his theory as being based on an alternative foundation
which takes pairs-of-sets as fundamental. But there's
no reason why you can't do it his way and still think
of what you're doing as working with sets.

> (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.

Not really. The question was just "Is Z a subset of R?",
and I bet the questioner would have been just as happy
to have said "Is Z a subclass of R?" but hadn't heard
that some mathematicians, for some purposes, like to
distinguish sets from classes. And, in any case, all
that the question supposes even if you insist on the
word "set" is that Z and R are sets, not that their
members are sets. (They might be sets of urelements
in a theory that allows such things.)

>                                                  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).)

The latter of those can perfectly well be an implementation
in terms of sets. Not if your set theory is ZF or NBG,
of course, but those are by no means the only options.

                             *

This has long ceased to have anything to do with Python.
I suggest that we take it to e-mail or drop it entirely.
(But if you prefer continuing in c.l.py, I don't mind.)

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