Turn off ZeroDivisionError?

[Mark Dickinson]

On Feb 16, 7:08 pm, Steven D'Aprano <st... at REMOVE-THIS-
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