[EMAIL PROTECTED] (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 ;-). -- http://mail.python.org/mailman/listinfo/python-list
