Terry Reedy <[EMAIL PROTECTED]> wrote: ... > I suspect that the notion of empty set was once controversial.
Yep: Reverend Dodgson (best known by his pen name of Lewis Carroll, and as the author of the Alice novels, but a logician and mathematician IRL) fought long and hard against Cantor's set theory, focusing on "the empty set" (singular: in Cantor's theory there can be only one) but not managing to build an actual contradiction around it. The poor man died just before his compatriot Bertrand Russell found "Russell's Paradox" (which destroyed Frege's logic -- but is quite applicable to destroying a foundation stone of Cantor's set theory as well). I personally believe that Dodgson was reaching towards what today we call modal logic (particularly intensional logic), though he could hardly get there with such encumbrances as his fiction writing, his photography, his heavy smoking of cannabis, etc, etc. But, that's just me, and I can't claim to be an expert at modern set theory (though I do know enough of it to see that it's quite different from Cantor's "naive" version that's still taught in most schools...!-), much less of the subtleties of modal logic. Still, if you give a set interpretation of modal logic, there isn't ONE empty set: the crucial point of modal logic, from my semi-learned POV, is that it distinguishes what JUST HAPPENS to be false, from what MUST INTRINSICALLY be false (and ditto for true). In set terms, say, "all integers x such that x>x" would be an *intrinsically* empty set, while "all dogs that are in this house right now" would be a set which *just happens* to be empty -- they aren't "one and the same, the sole empty set" any more than they are in commonsense (the notion Dodgson fought against). Alex -- http://mail.python.org/mailman/listinfo/python-list