So far AFAIK we have two kinds of infinity: Those that can be accommodated at the Grand Hilbert (e.g. integers, fractions, etc.) and those that cannot (set of irrational numbers, set of curves, set of polytopes, etc.) This was why we had to differentiate orders of infinity, e.g. ℵ₀ (The Grand Hilbert set), ℵ₁ (the irrational set, the real set), ℵ₂ (???), ℵ₃ (???), ℵₙ (???)
For values of infinity that are in higher orders than ℵ₀, we can only tell if they’re equal to ℵ₁ or undetermined, which means their infinity size is ℵ₁ or greater, but still unknown.
Unless someone did some Nobel prize worthy work in mathematics that I haven’t heard about which is quite possible.