Why is dictionary.keys() a list and not a set?

[Alex Martelli]

Christoph Zwerschke <cito at online.de> wrote:

> Alex Martelli wrote:
> 
> > An alternative theory, of course, is "God made the natural numbers; all
> > else is the work of man" -- and that one is by a German, too (Kronecker,
> > if I recall correctly).
> 
> Yes, it was Kronecker. But even natural numbers are usually constructed
> with sets using Peano's axioms.

Peano's axioms are perfectly abstract, as far as I recall.  Russell and
Whitehead did try to construct naturals from sets (defining, e.g., '5'
as "the set of all sets with five items"), but that was before the
inherent contradictions of set theory were widely known (though Russell
himself had destroyed Frege's attempts at theorization by pointing out
one such contradiction, the one wrt the "set of all sets that don't
include themselves as a member" if I recall correctly).  Later, Goedel
showed that any purely formal theory that's powerful enough to model
natural arithmetic cannot be both complete and consistent...


>  > The hope to found all of mathematics on set theory was primarily a
>  > _British_ effort, as I see it (Russell and Whitehead), and failed a
>  > long time ago... I'm not sure what, if anything, a mathematician of
>  > today would propose as the foundational theory...
> 
> Perhaps "string theory" ;-) So probably strings should become the 

Some physicists, maybe, surely not mathematicians though!


Alex