Is $\infty - \infty$ still indeterminate when the infinities are of different cardinality?

Question

$\infty - \infty$ is indeterminate.

Is $\aleph_1 - \aleph_0$ also indeterminate?

Answer 1

These are completely unrelated questions.

The notion of extended real number — the kind of thing which $\infty$ and $-\infty$ are — is geometric in nature; it has absolutely nothing to do with cardinality.

The reason $\aleph_0 - \aleph_0$ is undefined in cardinal number arithmetic is completely different than the reason $\infty - \infty$ is undefined in extended real number arithmetic.

In cardinal number arithmetic, it would be reasonable to define $\aleph_1 - \aleph_0 = \aleph_1$. Or more generally, if $\alpha$ and $\beta$ are cardinal numbers with $\alpha < \beta$ and $\beta$ infinite, then it would be reasonable to define $\beta - \alpha = \beta$.