[efggjrfp] infinity breaking groups

consider adding one infinite element to the integers or reals under addition and seeing if it remains a group.

first, consider defining 1 + infinity = infinity in the style of Cantor's cardinal numbers.

(1 + infinity) + (-infinity)
= infinity + (-infinity)
= 0

1 + (infinity + (-infinity))
= 1 + 0
= 1

associativity violated, so not a group.  note that, by group axioms, -infinity must exist, and -infinity + infinity = 0.

ordinal numbers are not a group because omega + (-1) is not defined.

the Riemann sphere (complex numbers extended with infinity) is not a group because infinity + (-infinity) is not defined.  (-infinity) itself might not be defined.

some ways to extend reals with infinity to get a group: Levi-Civita field (not too difficult to understand), surreal numbers (rather difficult to understand).  yet more exist, typically also difficult to understand.

the above successful ways get you not just a group but a field (with multiplication and division).  is that necessary?

unstated: why do you even want a group of numbers with infinity?