I found the following definition:
Definition. A set is countable iff its cardinality is either finite or
equal to $\aleph_0$. A set is denumerable iff its cardinality is exactly $\aleph_0$. A
set is uncountable iff its cardinality is greater than $\aleph_0$.The null set is countable. The finite set, {A, B, C}, is countable.
The infinite set, $\mathbb{N}$, is countable and denumerable. Sets with a larger cardinality than $\mathbb{N}$ are uncountable.
I have trouble with seeing the difference between countable and denumerable, apart from the part that the cardinality is finite. Isn't "A set is countable iff its cardinality equal to $\aleph_0$" and "A set is denumerable iff its cardinality is exactly $\aleph_0$" the same?
Best Answer
Every square is a rectangle, but not every rectangle is a square. Similarly, every denumerable set is countable, but not every countable set is denumerable. If you want, think of "denumerable" as an abbreviation for "countable and infinite" (or think of "countable" as an abbreviation for "denumerable or finite").