Turn off ZeroDivisionError?

[Mark Dickinson]

The real line, considered as a topological space, has limit points.
Two of them.

Ignore that. It was nonsense. A better statement: the completion (in
the sense of lattices) of the real numbers is (isomorphic to) the
doubly-extended real line. It's in this sense that +infinity and -
infinity can be considered limits.

I've no clue where your (Steven's) idea that 'all ordinals are surreal
numbers' came from. They're totally unrelated.

Sorry. I haven't had any dinner. I get tetchy when I haven't had any
dinner.

Usenet'ly yours,

Mark

 

[Steven D'Aprano]

I've no clue where your (Steven's) idea that 'all ordinals are surreal
numbers' came from. They're totally unrelated.

Tell that to John Conway.

Just as the *real* numbers fill in the gaps between the integers, the
*surreal* numbers fill in the gaps between Cantor's ordinal numbers. We
get them by generalizing our use of the {|} notation for the ordinal
numbers.
[...]
The ordinal numbers are those where there aren't any numbers to the right
of the bar:

{|} = 0, the simplest number of all
{0|} = 1, the simplest number greater than 0
{0,1|} = 2, the simplest number greater than 1 (and 0)

and so on.
[end quote]

"The Book of Numbers", John W Conway and Richard K Guy, Copernicus Books,
1996, p.283.

I trust I don't have to explain this to Mark, but for the benefit of
anyone else reading, Conway invented surreal numbers.

 

[Mark Dickinson]

Tell that to John Conway.

Apparently I also get stupid when I haven't had any dinner. Or
perhaps dinner has nothing to do with it. I was thinking of the
nonstandard reals.

You're absolutely right, and I hereby forfeit my Ph.D. (for the second
time today, as it happens).

Mark