I have commented on Steven's comments about alephs below.

It seems to me that this discussion (on having "different" infinities and allowing/storing arithmetic on them) is dead-on-arrival because:
- the scope of people who would find this useful is very small
- it would change current behaviour
- it would be unusual/a first among (popular) programming languages*
- consistency is basically impossible: as somebody pointed out, if you have a (Python) function that is 1 / (x ** 2), will the outcome be (1/0)**2 or just 1/0? (1/0)**2 is the consistent outcome, but would require implementing zeros with multiplicity...

This would also create problems underlying, because Python floats (I guess) correspond to the C and CPU floats, so performance and data structure/storage (have to store additional data beyond a C(PU) float to deal with infinities and their size, which would be unbounded if you e.g. allowed exponentials as you would get into things like Cantor Normal Form (https://en.wikipedia.org/wiki/Ordinal_arithmetic#Cantor_normal_form)). 

I honestly think this way lies madness for the floats of a general purpose programming language.

* of course, somebody has got to be first!

On Sun, 18 Oct 2020 at 23:03, Steven D'Aprano <steve@pearwood.info> wrote:
Oops, I messed up. (Thanks David for pointing that out.)

On Sun, Oct 18, 2020 at 07:45:40PM +1100, Steven D'Aprano wrote:

> Each of these number systems have related, but slightly different,
> rules. For example, IEEE-754 has a single signed infinity and 2**INF is
> exactly equal to INF. But in transfinite arithmetic, 2**INF is strictly
> greater than INF (for every infinity):
>     2**aleph_0 < aleph_1
>     2**aleph_1 < aleph_2
>     2**aleph_2 < aleph_3

I conflated what I was thinking:

    # note the change in comparison
    2**aleph_0 > aleph_0
    2**aleph_1 > aleph_1
    2**aleph_2 > aleph_2

which I think is correct regardless of your position on the Continuum
Hypothesis (David, care to comment?),

Yes, due to Cantor's diagonal argument: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument  

with this:

    2**aleph_0 = aleph_1
    2**aleph_1 = aleph_2
    2**aleph_2 = aleph_3

which is only true if the Continuum Hypothesis is true

*generalized* Continuum Hypothesis, and in fact it isn't "only true if (G)CH" is much stronger, it is in fact the same statement (if your "..." means "for all ordinals" as GCH (CH is just the first one of those). See https://en.wikipedia.org/wiki/Continuum_hypothesis#The_generalized_continuum_hypothesis.