🚀 go-pugleaf

jad1002@hermes.cam.ac.uk  wrote on 23.10.95 in <jad1002.122.000AD3E8@hermes.cam.ac.uk>:

> In article <5wJmz3sjcsB@khms.westfalen.de> kai@khms.westfalen.de (Kai
> >Henningsen) writes: Subject: Re: Strings, storage management, overloading

> >However, when you speak about "the set of the natural numbers", then you
> >speak about the same thing, regardless of which representation you are
> >thinking of - or else you have to give it a different name.

[...]

> >And this thing *is* a subset of the real numbers. There is *no* technical
> >difference to be found. (Or else point one out!)
>
> I'll accept that challenge.
>
> The difference lies in the _construction_ of these entities. In some sense,

Umm ... you can, as has been noted, construct these several ways. So, the  
construction cannot be a fixed attribute of "the numbers themselves". That  
makes arguments by construction somewhat useless. (Or else you get  
different natural number 2's depending on how you construct them. In that  
sense, of course reals are different again; however, I don't think that's  
the sense we're after here :-))

> So, I argue both ways... yes, something with every property we require of
> the natural numbers is a subset of the real numbers, but in certain
> technical areas it is convenient or essential to separate the two notions.

There's still nothing that makes all "natural" 2's the same, and all  
"real" 2's the same, but still makes these two concepts different.

> >Aha! There is the error. The sets we are speaking about can be constructed
> >in a number of ways; however, the sets themselves are the same however we
> >derive them.
>
> Urm, no. See the above. This is a set-theoretic debate.

See above. Once you make it useful to speak about "the natural number 2",  
you make it impossible to have "the real number 2" be something different.

> >Yeah, yeah. "Almost all" is, after all, mathematical slang for "all except
> >a finite number", and there is only a finite number of elements computer
> >integers or reals can have (except, maybe, LISP bignums :-)).
>
> Actually, this is wrong too, and here I argue purely because I'm feeling

True, this should be a countable number. Anyway, that was only an aside  
:-)

Anyway, this is starting to get too far away from programming languages.  
We should either take this to some other group or to email.

Kai
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