David Mertz writes:
On Sun, Oct 11, 2020, 9:07 PM Steven D'Aprano
Even in the cardinal numbers, two times infinity (aleph-0) is just aleph-0; however you might be pleased to know that two to the power of aleph-0 is aleph-1.
Oh... So you're one of those who believe the Continuum Hypothesis :-).
As any good defense lawyer can tell you, you don't need to believe something to use it to prove something else. :-)
I was going to mention Surreal Numbers also, but I see you touched on it.
As far as what Steven discussed, the ordinal numbers have the same properties (except I've never heard of ω-1 in a discussion of ordinals, but it should work I think). (Maybe the surreals are constructed from the ordinals as the reals are constrcuted from the cardinals?)