There are more possible cardinalities above the the Cardinality of the continuum which is $2^{\aleph_0}$ (e.g. $2^{2^{\aleph_0}}$ - that wikipedia page even mentions this, have you actually read that), sets of those sizes generally don't contain discrete things, so your distinction is simply wrong, but at lot of it lies in equalling "principally countable" with "doesn't have the Cardinality of the continuum property".
In a lot of cases it doesn't make any sense to label something as either discrete or continuous, even giving the word "continuous" meaning probably requires some metric.
My take on a criterion to differentiate would be whether every object has a next object (as a criterion for discreteness), and that leads to a countable (perhaps finite) set, whereas continuous means there's an object between (that's where the existence of a metric comes in) any two objects in the set.
Grove
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2021-08-19T14:45:30Z (over 2 years ago)