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

[David C. Ullrich]

On Sat, 28 Jul 2001 19:34:46 +0100, Gareth.McCaughan at pobox.com (Gareth
McCaughan) wrote:

>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,

That's one I mentioned. With that approach 2 is not a set.
(Not a set in the currently very very standard set theory.)

> 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).

That's a good example, because it seems much more
natural than the "standard" implementation. Takes
a while, possibly infintitely long, to see what's
better about the vN version.

>This has long ceased to have anything to do with Python.
>I suggest that we take it to e-mail or drop it entirely.

Simply dropping it seems the sensible option, since
neither of us is actually saying anything the other
doesn't know - my original butting in was a just for
the record thing.

(Did they change the rules recently? Don't recall
relevance to Python being a criterion here...)

>(But if you prefer continuing in c.l.py, I don't mind.)


David C. Ullrich