Comments on What is "continuous" in Math?
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What is "continuous" in Math?
I understand that in Math, there is a common separation between discrete and continuous.
- I understand that in Math we could say that discrete is anything (any set?) which is principally countable;
I therefore assume that in Math, "continuous" would be the opposite of discrete, hence anything (any set?) which is principally noncountable, but it might be wrong.
What is "continuous" in Math?
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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.
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