Different types of uncountability
Do there exist different types of uncountability? I am well aware that
$o(\aleph_0) < o (\mathbb{R}) < o (\mathcal{P}(\mathbb{R}))$, and so on.
I have looked around to no avail, however; is there a stricter definition
for cardinality than "two cardinal numbers are the same if there exists a
1-1 correspondence between their sets"? That is, is there a rigorous
definition for uncountability?
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