Why is dictionary.keys() a list and not a set?
Paul Rubin
http
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.
http://en.wikipedia.org/wiki/Naive_set_theory
http://en.wikipedia.org/wiki/Axiomatic_set_theory
> Later, Goedel showed that any purely formal theory that's powerful
> enough to model natural arithmetic cannot be both complete and
> consistent...
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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