Do all the infinite sets have the same cardinality?
No. There are cardinalities strictly greater than |N|. In particular,
|N|<|R| (there are “more” real numbers than natural numbers).
Sets that are finite or have the same cardinality as N (or Z+, the set of
positive integers) are called countable.
Countable sets are sets whose elements can be listed:
(if the set is finite) or
(if the set is infinite).