Meaning of countably infinite | Babel Free

Definitions

That is both countable and infinite; having the same cardinality as the set of natural numbers; formally, such that a bijection exists from ℕ to the set.

not-comparable

Examples

“1953 [Addison-Wesley], Bruce Elwyn Meserve, Fundamental Concepts of Algebra, 1982, Dover, page 36, This one-to-one correspondence between the set⟳ of positive integers and the set⟳ of pairs of positive integers indicates that the set⟳ of pairs is countably infinite. Since the set⟳ of positive rational numbers is a subset of the set⟳ of all pairs of positive integers, the set⟳ of positive rational numbers is at most countably infinite. Then, since it is also at least countably infinite, the set⟳ of positive rational numbers is countably infinite.”

“Instead of saying that the aggregate of natural numbers is countably infinite (see⟳ Chapter 24), one can use⟳ Cantor's symbolism and state that its cardinal number⟳ is #92;aleph#95;0 (read⟳ aleph null).[…]Now the range of a random variable may be a finite set⟳, a countably infinite set⟳, or a continuum.”

“The term “countably infinite” would seem to suggest⟳ that such a set⟳ is infinite. However, the definitions of “countably infinite” and “infinite” were made separately, and so we have⟳ to prove⟳ that countably infinite sets are indeed infinite (otherwise our notation would be rather misleading).”

CEFR level