Why is dictionary.keys() a list and not a set?
Thu Nov 24 23:33:19 CET 2005
aleax at mail.comcast.net (Alex Martelli) writes:
> 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).
That's only what's called "naive set theory". Axiomatic set theory
(the kind they use now) doesn't have these issues.
> Later, Goedel showed that any purely formal theory that's powerful
> enough to model natural arithmetic cannot be both complete and
Yes, it's not considered terribly troublesome. Set theory (and
arithmetic) are generally accepted as being consistent but incomplete.
So there will always be something for mathemeticians to do ;-).
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