a set is said to be countably infinite (or just countable) iff it has equal cardinality with the natural numbers . a set is said to be at most countable iff it is either countable or finite. we say that a set is uncountable if it is infinite but not countable.
the same definition can be found in (in Jay Cummings, 2019 chapter 2 cardinality definition 2.10).