From the book $\mathbf{A~Readable~Introduction~to~Real~Mathematics}$ by Daniel, David, Peter Rosenthal.
Definition 10.2.1. A set is countable (sometimes called denumerable, or enumerable) if it is either finite or has the same cardinality as the set of natural numbers. A set is said to be uncountable if it is not countable.
but I don't know wheather the statement if a set is infinite and does not have same cardinality as the set of natural numbers, then the set is uncountable holds as well ?
I know the Definition can not imply the statement I give. But I am just quite curious on the truth of the statement I provide above.
Yes an infinite set that has a cardinality that is not equal to the natural numbers is uncountable.