On Feb 16, 7:08 pm, Steven D'Aprano <[EMAIL PROTECTED] cybersource.com.au> wrote: > On Fri, 15 Feb 2008 17:31:51 -0800, Mark Dickinson wrote: > > Not sure that alephs have anything to do with it. And unless I'm > > missing something, minus aleph(0) is nonsense. (How do you define the > > negation of a cardinal?) > > *shrug* How would you like to?
> Since we have generalized the natural numbers to the integers > > ... -3 -2 -1 0 1 2 3 ... > > without worrying about what set has cardinality -1, I see no reason why > we shouldn't generalize negation to the alephs. The reason is that it doesn't give a useful result. There's a natural process for turning a commutative monoid into a group (it's the adjoint to the forgetful functor from groups to commutative monoids). Apply it to the "set of cardinals", leaving aside the set-theoretic difficulties with the idea of the "set of cardinals" in the first place, and you get the trivial group. > There's lots of hand-waving there. I expect a real mathematician could > make it all vigorous. Rigorous? Yes, I expect I could. And surreal numbers are something entirely different again. > That's a very informal definition of infinity. Taken literally, it's also > nonsense, since the real number line has no limit, so talking about the > limit of something with no limit is meaningless. So we have to take it > loosely. The real line, considered as a topological space, has limit points. Two of them. Mark -- http://mail.python.org/mailman/listinfo/python-list
