definition: countably infinite set
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.
(taken from terence tao, 2022 definition 8.1.1)
the same definition can be found in (in Jay Cummings, 2019 chapter 2 cardinality definition 2.10).